{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0001","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":1,\"b\":-1,\"x\":-12,\"y\":-3}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([-12, -3], [1, -1], -9, 否)","explanation_zh":"Wave5根節點核對=([-12, -3], [1, -1], -9, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011001,"parameters":{"a":1,"b":-1,"x":-12,"y":-3,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":1,\"b\":-1,\"x\":-12,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0002","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":2,\"b\":-2,\"x\":-11,\"y\":-2}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([-11, -2], [2, -2], -18, 否)","explanation_zh":"Wave5根節點核對=([-11, -2], [2, -2], -18, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011002,"parameters":{"a":2,"b":-2,"x":-11,"y":-2,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":2,\"b\":-2,\"x\":-11,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0003","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":3,\"b\":-3,\"x\":-10,\"y\":-1}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([-10, -1], [3, -3], -27, 否)","explanation_zh":"Wave5根節點核對=([-10, -1], [3, -3], -27, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011003,"parameters":{"a":3,"b":-3,"x":-10,"y":-1,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":3,\"b\":-3,\"x\":-10,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0004","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":4,\"b\":-4,\"x\":-9,\"y\":0}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([-9, 0], [4, -4], -36, 否)","explanation_zh":"Wave5根節點核對=([-9, 0], [4, -4], -36, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011004,"parameters":{"a":4,"b":-4,"x":-9,"y":0,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":4,\"b\":-4,\"x\":-9,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0005","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":5,\"b\":-5,\"x\":-8,\"y\":1}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([-8, 1], [5, -5], -45, 否)","explanation_zh":"Wave5根節點核對=([-8, 1], [5, -5], -45, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011005,"parameters":{"a":5,"b":-5,"x":-8,"y":1,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":5,\"b\":-5,\"x\":-8,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0006","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":6,\"b\":-1,\"x\":-7,\"y\":2}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([-7, 2], [6, -1], -44, 否)","explanation_zh":"Wave5根節點核對=([-7, 2], [6, -1], -44, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011006,"parameters":{"a":6,"b":-1,"x":-7,"y":2,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":6,\"b\":-1,\"x\":-7,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0007","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":7,\"b\":-2,\"x\":-6,\"y\":3}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([-6, 3], [7, -2], -48, 否)","explanation_zh":"Wave5根節點核對=([-6, 3], [7, -2], -48, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011007,"parameters":{"a":7,"b":-2,"x":-6,"y":3,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":7,\"b\":-2,\"x\":-6,\"y\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0008","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":8,\"b\":-3,\"x\":-5,\"y\":-3}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([-5, -3], [8, -3], -31, 否)","explanation_zh":"Wave5根節點核對=([-5, -3], [8, -3], -31, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011008,"parameters":{"a":8,"b":-3,"x":-5,"y":-3,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":8,\"b\":-3,\"x\":-5,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0009","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":9,\"b\":-4,\"x\":-4,\"y\":-2}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([-4, -2], [9, -4], -28, 否)","explanation_zh":"Wave5根節點核對=([-4, -2], [9, -4], -28, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011009,"parameters":{"a":9,"b":-4,"x":-4,"y":-2,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":9,\"b\":-4,\"x\":-4,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0010","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":10,\"b\":-5,\"x\":-3,\"y\":-1}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([-3, -1], [10, -5], -25, 否)","explanation_zh":"Wave5根節點核對=([-3, -1], [10, -5], -25, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011010,"parameters":{"a":10,"b":-5,"x":-3,"y":-1,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":10,\"b\":-5,\"x\":-3,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0011","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":11,\"b\":-1,\"x\":-2,\"y\":0}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([-2, 0], [11, -1], -22, 否)","explanation_zh":"Wave5根節點核對=([-2, 0], [11, -1], -22, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011011,"parameters":{"a":11,"b":-1,"x":-2,"y":0,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":11,\"b\":-1,\"x\":-2,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0012","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":12,\"b\":-2,\"x\":-1,\"y\":1}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([-1, 1], [12, -2], -14, 否)","explanation_zh":"Wave5根節點核對=([-1, 1], [12, -2], -14, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011012,"parameters":{"a":12,"b":-2,"x":-1,"y":1,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":12,\"b\":-2,\"x\":-1,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0013","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":13,\"b\":-3,\"x\":0,\"y\":2}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [13, -3], -6, 否)","explanation_zh":"Wave5根節點核對=([0, 2], [13, -3], -6, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011013,"parameters":{"a":13,"b":-3,"x":0,"y":2,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":13,\"b\":-3,\"x\":0,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0014","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":14,\"b\":-4,\"x\":1,\"y\":3}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([1, 3], [14, -4], 2, 否)","explanation_zh":"Wave5根節點核對=([1, 3], [14, -4], 2, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011014,"parameters":{"a":14,"b":-4,"x":1,"y":3,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":14,\"b\":-4,\"x\":1,\"y\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0015","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":15,\"b\":-5,\"x\":2,\"y\":-3}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([2, -3], [15, -5], 45, 否)","explanation_zh":"Wave5根節點核對=([2, -3], [15, -5], 45, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011015,"parameters":{"a":15,"b":-5,"x":2,"y":-3,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":15,\"b\":-5,\"x\":2,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0016","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":16,\"b\":-1,\"x\":3,\"y\":-2}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([3, -2], [16, -1], 50, 否)","explanation_zh":"Wave5根節點核對=([3, -2], [16, -1], 50, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011016,"parameters":{"a":16,"b":-1,"x":3,"y":-2,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":16,\"b\":-1,\"x\":3,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0017","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":17,\"b\":-2,\"x\":4,\"y\":-1}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([4, -1], [17, -2], 70, 否)","explanation_zh":"Wave5根節點核對=([4, -1], [17, -2], 70, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011017,"parameters":{"a":17,"b":-2,"x":4,"y":-1,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":17,\"b\":-2,\"x\":4,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0018","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":18,\"b\":-3,\"x\":5,\"y\":0}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([5, 0], [18, -3], 90, 否)","explanation_zh":"Wave5根節點核對=([5, 0], [18, -3], 90, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011018,"parameters":{"a":18,"b":-3,"x":5,"y":0,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":18,\"b\":-3,\"x\":5,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0019","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":19,\"b\":-4,\"x\":6,\"y\":1}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([6, 1], [19, -4], 110, 否)","explanation_zh":"Wave5根節點核對=([6, 1], [19, -4], 110, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011019,"parameters":{"a":19,"b":-4,"x":6,"y":1,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":19,\"b\":-4,\"x\":6,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0020","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":20,\"b\":-5,\"x\":7,\"y\":2}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([7, 2], [20, -5], 130, 否)","explanation_zh":"Wave5根節點核對=([7, 2], [20, -5], 130, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011020,"parameters":{"a":20,"b":-5,"x":7,"y":2,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":20,\"b\":-5,\"x\":7,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0021","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":21,\"b\":-1,\"x\":8,\"y\":3}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([8, 3], [21, -1], 165, 否)","explanation_zh":"Wave5根節點核對=([8, 3], [21, -1], 165, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011021,"parameters":{"a":21,"b":-1,"x":8,"y":3,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":21,\"b\":-1,\"x\":8,\"y\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0022","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":22,\"b\":-2,\"x\":9,\"y\":-3}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([9, -3], [22, -2], 204, 否)","explanation_zh":"Wave5根節點核對=([9, -3], [22, -2], 204, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011022,"parameters":{"a":22,"b":-2,"x":9,"y":-3,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":22,\"b\":-2,\"x\":9,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0023","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":23,\"b\":-3,\"x\":10,\"y\":-2}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([10, -2], [23, -3], 236, 否)","explanation_zh":"Wave5根節點核對=([10, -2], [23, -3], 236, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011023,"parameters":{"a":23,"b":-3,"x":10,"y":-2,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":23,\"b\":-3,\"x\":10,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0024","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":24,\"b\":-4,\"x\":11,\"y\":-1}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([11, -1], [24, -4], 268, 否)","explanation_zh":"Wave5根節點核對=([11, -1], [24, -4], 268, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011024,"parameters":{"a":24,"b":-4,"x":11,"y":-1,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":24,\"b\":-4,\"x\":11,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0025","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-line-membership；資料={\"a\":25,\"b\":-5,\"x\":12,\"y\":0}。請依 [point,normal,equation_value,in_line] 欄序回答。","answer":{"kind":"expression","value":"([12, 0], [25, -5], 300, 否)","explanation_zh":"Wave5根節點核對=([12, 0], [25, -5], 300, 否)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-line-membership","seed":202611011025,"parameters":{"a":25,"b":-5,"x":12,"y":0,"domain":"Exact_R2_subspace_certificate","operation":"subspace-line-membership","semantic_template":"qf-mod32-subspace-line-membership/{\"a\":25,\"b\":-5,\"x\":12,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0026","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":1,\"b\":-1}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, -1], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([1, -1], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011026,"parameters":{"a":1,"b":-1,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":1,\"b\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0027","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":2,\"b\":-2}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([2, -2], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([2, -2], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011027,"parameters":{"a":2,"b":-2,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":2,\"b\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0028","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":3,\"b\":-3}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([3, -3], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([3, -3], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011028,"parameters":{"a":3,"b":-3,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":3,\"b\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0029","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":4,\"b\":-4}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([4, -4], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([4, -4], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011029,"parameters":{"a":4,"b":-4,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":4,\"b\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0030","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":5,\"b\":-5}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([5, -5], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([5, -5], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011030,"parameters":{"a":5,"b":-5,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":5,\"b\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0031","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":6,\"b\":-6}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([6, -6], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([6, -6], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011031,"parameters":{"a":6,"b":-6,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":6,\"b\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0032","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":7,\"b\":-7}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([7, -7], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([7, -7], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011032,"parameters":{"a":7,"b":-7,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":7,\"b\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0033","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":8,\"b\":-1}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([8, -1], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([8, -1], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011033,"parameters":{"a":8,"b":-1,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":8,\"b\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0034","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":9,\"b\":-2}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([9, -2], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([9, -2], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011034,"parameters":{"a":9,"b":-2,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":9,\"b\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0035","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":10,\"b\":-3}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([10, -3], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([10, -3], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011035,"parameters":{"a":10,"b":-3,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":10,\"b\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0036","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":11,\"b\":-4}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([11, -4], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([11, -4], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011036,"parameters":{"a":11,"b":-4,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":11,\"b\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0037","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":12,\"b\":-5}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([12, -5], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([12, -5], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011037,"parameters":{"a":12,"b":-5,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":12,\"b\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0038","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":13,\"b\":-6}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([13, -6], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([13, -6], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011038,"parameters":{"a":13,"b":-6,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":13,\"b\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0039","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":14,\"b\":-7}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([14, -7], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([14, -7], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011039,"parameters":{"a":14,"b":-7,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":14,\"b\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0040","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":15,\"b\":-1}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([15, -1], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([15, -1], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011040,"parameters":{"a":15,"b":-1,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":15,\"b\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0041","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":16,\"b\":-2}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([16, -2], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([16, -2], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011041,"parameters":{"a":16,"b":-2,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":16,\"b\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0042","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":17,\"b\":-3}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([17, -3], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([17, -3], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011042,"parameters":{"a":17,"b":-3,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":17,\"b\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0043","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":18,\"b\":-4}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([18, -4], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([18, -4], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011043,"parameters":{"a":18,"b":-4,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":18,\"b\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0044","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":19,\"b\":-5}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([19, -5], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([19, -5], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011044,"parameters":{"a":19,"b":-5,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":19,\"b\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0045","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":20,\"b\":-6}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([20, -6], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([20, -6], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011045,"parameters":{"a":20,"b":-6,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":20,\"b\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0046","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":21,\"b\":-7}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([21, -7], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([21, -7], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011046,"parameters":{"a":21,"b":-7,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":21,\"b\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0047","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":22,\"b\":-1}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([22, -1], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([22, -1], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011047,"parameters":{"a":22,"b":-1,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":22,\"b\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0048","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":23,\"b\":-2}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([23, -2], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([23, -2], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011048,"parameters":{"a":23,"b":-2,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":23,\"b\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0049","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":24,\"b\":-3}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([24, -3], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([24, -3], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011049,"parameters":{"a":24,"b":-3,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":24,\"b\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0050","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-universal-line-certificate；資料={\"a\":25,\"b\":-4}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。","answer":{"kind":"expression","value":"([25, -4], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)","explanation_zh":"Wave5根節點核對=([25, -4], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-universal-line-certificate","seed":202611011050,"parameters":{"a":25,"b":-4,"domain":"Exact_R2_subspace_certificate","operation":"subspace-universal-line-certificate","semantic_template":"qf-mod32-subspace-universal-line-certificate/{\"a\":25,\"b\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0051","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[1,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [1, 1]], [scalar_failure, 2, [1, 1], [2, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [1, 1]], [scalar_failure, 2, [1, 1], [2, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011051,"parameters":{"points":[[0,0],[1,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0052","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[2,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [2, 1]], [scalar_failure, 2, [2, 1], [4, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [2, 1]], [scalar_failure, 2, [2, 1], [4, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011052,"parameters":{"points":[[0,0],[2,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0053","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[3,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [3, 1]], [scalar_failure, 2, [3, 1], [6, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [3, 1]], [scalar_failure, 2, [3, 1], [6, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011053,"parameters":{"points":[[0,0],[3,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0054","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[4,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [4, 1]], [scalar_failure, 2, [4, 1], [8, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [4, 1]], [scalar_failure, 2, [4, 1], [8, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011054,"parameters":{"points":[[0,0],[4,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0055","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[5,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [5, 1]], [scalar_failure, 2, [5, 1], [10, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [5, 1]], [scalar_failure, 2, [5, 1], [10, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011055,"parameters":{"points":[[0,0],[5,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0056","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[6,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [6, 1]], [scalar_failure, 2, [6, 1], [12, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [6, 1]], [scalar_failure, 2, [6, 1], [12, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011056,"parameters":{"points":[[0,0],[6,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[6,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0057","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[7,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [7, 1]], [scalar_failure, 2, [7, 1], [14, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [7, 1]], [scalar_failure, 2, [7, 1], [14, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011057,"parameters":{"points":[[0,0],[7,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[7,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0058","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[8,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [8, 1]], [scalar_failure, 2, [8, 1], [16, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [8, 1]], [scalar_failure, 2, [8, 1], [16, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011058,"parameters":{"points":[[0,0],[8,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[8,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0059","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[9,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [9, 1]], [scalar_failure, 2, [9, 1], [18, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [9, 1]], [scalar_failure, 2, [9, 1], [18, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011059,"parameters":{"points":[[0,0],[9,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[9,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0060","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[10,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [10, 1]], [scalar_failure, 2, [10, 1], [20, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [10, 1]], [scalar_failure, 2, [10, 1], [20, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011060,"parameters":{"points":[[0,0],[10,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[10,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0061","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[11,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [11, 1]], [scalar_failure, 2, [11, 1], [22, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [11, 1]], [scalar_failure, 2, [11, 1], [22, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011061,"parameters":{"points":[[0,0],[11,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[11,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0062","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[12,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [12, 1]], [scalar_failure, 2, [12, 1], [24, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [12, 1]], [scalar_failure, 2, [12, 1], [24, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011062,"parameters":{"points":[[0,0],[12,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[12,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0063","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[13,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [13, 1]], [scalar_failure, 2, [13, 1], [26, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [13, 1]], [scalar_failure, 2, [13, 1], [26, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011063,"parameters":{"points":[[0,0],[13,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[13,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0064","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[14,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [14, 1]], [scalar_failure, 2, [14, 1], [28, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [14, 1]], [scalar_failure, 2, [14, 1], [28, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011064,"parameters":{"points":[[0,0],[14,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[14,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0065","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[15,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [15, 1]], [scalar_failure, 2, [15, 1], [30, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [15, 1]], [scalar_failure, 2, [15, 1], [30, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011065,"parameters":{"points":[[0,0],[15,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[15,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0066","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[16,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [16, 1]], [scalar_failure, 2, [16, 1], [32, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [16, 1]], [scalar_failure, 2, [16, 1], [32, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011066,"parameters":{"points":[[0,0],[16,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[16,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0067","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[17,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [17, 1]], [scalar_failure, 2, [17, 1], [34, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [17, 1]], [scalar_failure, 2, [17, 1], [34, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011067,"parameters":{"points":[[0,0],[17,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[17,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0068","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[18,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [18, 1]], [scalar_failure, 2, [18, 1], [36, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [18, 1]], [scalar_failure, 2, [18, 1], [36, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011068,"parameters":{"points":[[0,0],[18,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[18,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0069","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[19,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [19, 1]], [scalar_failure, 2, [19, 1], [38, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [19, 1]], [scalar_failure, 2, [19, 1], [38, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011069,"parameters":{"points":[[0,0],[19,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[19,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0070","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[20,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [20, 1]], [scalar_failure, 2, [20, 1], [40, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [20, 1]], [scalar_failure, 2, [20, 1], [40, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011070,"parameters":{"points":[[0,0],[20,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[20,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0071","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[21,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [21, 1]], [scalar_failure, 2, [21, 1], [42, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [21, 1]], [scalar_failure, 2, [21, 1], [42, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011071,"parameters":{"points":[[0,0],[21,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[21,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0072","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[22,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [22, 1]], [scalar_failure, 2, [22, 1], [44, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [22, 1]], [scalar_failure, 2, [22, 1], [44, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011072,"parameters":{"points":[[0,0],[22,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[22,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0073","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[23,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [23, 1]], [scalar_failure, 2, [23, 1], [46, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [23, 1]], [scalar_failure, 2, [23, 1], [46, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011073,"parameters":{"points":[[0,0],[23,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[23,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0074","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[24,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [24, 1]], [scalar_failure, 2, [24, 1], [48, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [24, 1]], [scalar_failure, 2, [24, 1], [48, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011074,"parameters":{"points":[[0,0],[24,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[24,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0075","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[25,1]]}。請依 [points,witness,subspace,classification] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [25, 1]], [scalar_failure, 2, [25, 1], [50, 2]], 否, not_subspace_exact_counterexample)","explanation_zh":"Wave5根節點核對=([[0, 0], [25, 1]], [scalar_failure, 2, [25, 1], [50, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-finite-set-counterexample","seed":202611011075,"parameters":{"points":[[0,0],[25,1]],"domain":"Exact_R2_subspace_certificate","operation":"subspace-finite-set-counterexample","semantic_template":"qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[25,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0076","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":0,\"b\":1,\"c\":-12}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, -12], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([0, 1, -12], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011076,"parameters":{"a":0,"b":1,"c":-12,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":0,\"b\":1,\"c\":-12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0077","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":1,\"b\":2,\"c\":-11}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, -11], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([1, 2, -11], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011077,"parameters":{"a":1,"b":2,"c":-11,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":1,\"b\":2,\"c\":-11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0078","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":2,\"b\":3,\"c\":-10}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([2, 3, -10], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([2, 3, -10], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011078,"parameters":{"a":2,"b":3,"c":-10,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":2,\"b\":3,\"c\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0079","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":3,\"b\":4,\"c\":-9}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([3, 4, -9], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([3, 4, -9], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011079,"parameters":{"a":3,"b":4,"c":-9,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":3,\"b\":4,\"c\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0080","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":4,\"b\":5,\"c\":-8}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([4, 5, -8], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([4, 5, -8], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011080,"parameters":{"a":4,"b":5,"c":-8,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":4,\"b\":5,\"c\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0081","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":0,\"b\":6,\"c\":-7}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 6, -7], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([0, 6, -7], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011081,"parameters":{"a":0,"b":6,"c":-7,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":0,\"b\":6,\"c\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0082","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":1,\"b\":0,\"c\":-6}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, -6], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([1, 0, -6], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011082,"parameters":{"a":1,"b":0,"c":-6,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":1,\"b\":0,\"c\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0083","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":2,\"b\":1,\"c\":-5}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([2, 1, -5], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([2, 1, -5], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011083,"parameters":{"a":2,"b":1,"c":-5,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":2,\"b\":1,\"c\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0084","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":3,\"b\":2,\"c\":-4}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([3, 2, -4], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([3, 2, -4], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011084,"parameters":{"a":3,"b":2,"c":-4,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":3,\"b\":2,\"c\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0085","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":4,\"b\":3,\"c\":-3}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([4, 3, -3], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([4, 3, -3], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011085,"parameters":{"a":4,"b":3,"c":-3,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":4,\"b\":3,\"c\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0086","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":0,\"b\":4,\"c\":-2}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 4, -2], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([0, 4, -2], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011086,"parameters":{"a":0,"b":4,"c":-2,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":0,\"b\":4,\"c\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0087","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":1,\"b\":5,\"c\":-1}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 5, -1], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([1, 5, -1], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011087,"parameters":{"a":1,"b":5,"c":-1,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":1,\"b\":5,\"c\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0088","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":2,\"b\":6,\"c\":0}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([2, 6, 0], homogeneous_line, 是, universal_homogeneous_equation_certificate)","explanation_zh":"Wave5根節點核對=([2, 6, 0], homogeneous_line, 是, universal_homogeneous_equation_certificate)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011088,"parameters":{"a":2,"b":6,"c":0,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":2,\"b\":6,\"c\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0089","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":3,\"b\":0,\"c\":1}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 1], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([3, 0, 1], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011089,"parameters":{"a":3,"b":0,"c":1,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":3,\"b\":0,\"c\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0090","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":4,\"b\":1,\"c\":2}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([4, 1, 2], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([4, 1, 2], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011090,"parameters":{"a":4,"b":1,"c":2,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":4,\"b\":1,\"c\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0091","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":0,\"b\":2,\"c\":3}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 3], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([0, 2, 3], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011091,"parameters":{"a":0,"b":2,"c":3,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":0,\"b\":2,\"c\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0092","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":1,\"b\":3,\"c\":4}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 4], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([1, 3, 4], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011092,"parameters":{"a":1,"b":3,"c":4,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":1,\"b\":3,\"c\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0093","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":2,\"b\":4,\"c\":5}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([2, 4, 5], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([2, 4, 5], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011093,"parameters":{"a":2,"b":4,"c":5,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":2,\"b\":4,\"c\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0094","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":3,\"b\":5,\"c\":6}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 6], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([3, 5, 6], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011094,"parameters":{"a":3,"b":5,"c":6,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":3,\"b\":5,\"c\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0095","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":4,\"b\":6,\"c\":7}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([4, 6, 7], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([4, 6, 7], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011095,"parameters":{"a":4,"b":6,"c":7,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":4,\"b\":6,\"c\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0096","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":0,\"b\":0,\"c\":8}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 8], empty_set, 否, empty_set)","explanation_zh":"Wave5根節點核對=([0, 0, 8], empty_set, 否, empty_set)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011096,"parameters":{"a":0,"b":0,"c":8,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":0,\"b\":0,\"c\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0097","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":1,\"b\":1,\"c\":9}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 9], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([1, 1, 9], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011097,"parameters":{"a":1,"b":1,"c":9,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":1,\"b\":1,\"c\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0098","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":2,\"b\":2,\"c\":10}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 10], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([2, 2, 10], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011098,"parameters":{"a":2,"b":2,"c":10,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":2,\"b\":2,\"c\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0099","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":3,\"b\":3,\"c\":11}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([3, 3, 11], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([3, 3, 11], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011099,"parameters":{"a":3,"b":3,"c":11,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":3,\"b\":3,\"c\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r01-vector-subspace-0100","batch_id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-subspace"],"title_zh":"實向量子空間","stem_zh":"R²通常運算與明示全稱代數證書。操作=subspace-classification；資料={\"a\":4,\"b\":4,\"c\":12}。請依 [equation,classification,subspace,certificate] 欄序回答。","answer":{"kind":"expression","value":"([4, 4, 12], affine_line_not_through_zero, 否, affine_line_not_through_zero)","explanation_zh":"Wave5根節點核對=([4, 4, 12], affine_line_not_through_zero, 否, affine_line_not_through_zero)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實向量子空間"]},"variant":{"family_id":"qf-mod32-subspace-classification","seed":202611011100,"parameters":{"a":4,"b":4,"c":12,"domain":"Exact_R2_subspace_certificate","operation":"subspace-classification","semantic_template":"qf-mod32-subspace-classification/{\"a\":4,\"b\":4,\"c\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}