{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0001","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[-12,-3,2],\"right\":[-2,-8,0]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([-12, -3, 2], [-2, -8, 0], [-14, -11, 2])","explanation_zh":"結構基礎核對=([-12, -3, 2], [-2, -8, 0], [-14, -11, 2])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224001,"parameters":{"left":[-12,-3,2],"right":[-2,-8,0],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-12,-3,2],\"right\":[-2,-8,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0002","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[-11,-2,1],\"right\":[-1,-7,1]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([-11, -2, 1], [-1, -7, 1], [-12, -9, 2])","explanation_zh":"結構基礎核對=([-11, -2, 1], [-1, -7, 1], [-12, -9, 2])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224002,"parameters":{"left":[-11,-2,1],"right":[-1,-7,1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-11,-2,1],\"right\":[-1,-7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0003","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[-10,-1,0],\"right\":[0,-6,2]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([-10, -1, 0], [0, -6, 2], [-10, -7, 2])","explanation_zh":"結構基礎核對=([-10, -1, 0], [0, -6, 2], [-10, -7, 2])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224003,"parameters":{"left":[-10,-1,0],"right":[0,-6,2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-10,-1,0],\"right\":[0,-6,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0004","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[-9,0,-1],\"right\":[1,-5,0]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([-9, 0, -1], [1, -5, 0], [-8, -5, -1])","explanation_zh":"結構基礎核對=([-9, 0, -1], [1, -5, 0], [-8, -5, -1])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224004,"parameters":{"left":[-9,0,-1],"right":[1,-5,0],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-9,0,-1],\"right\":[1,-5,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0005","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[-8,1,-2],\"right\":[2,-4,1]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([-8, 1, -2], [2, -4, 1], [-6, -3, -1])","explanation_zh":"結構基礎核對=([-8, 1, -2], [2, -4, 1], [-6, -3, -1])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224005,"parameters":{"left":[-8,1,-2],"right":[2,-4,1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-8,1,-2],\"right\":[2,-4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0006","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[-7,2,-3],\"right\":[-2,-3,2]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([-7, 2, -3], [-2, -3, 2], [-9, -1, -1])","explanation_zh":"結構基礎核對=([-7, 2, -3], [-2, -3, 2], [-9, -1, -1])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224006,"parameters":{"left":[-7,2,-3],"right":[-2,-3,2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-7,2,-3],\"right\":[-2,-3,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0007","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[-6,3,-4],\"right\":[-1,-2,0]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([-6, 3, -4], [-1, -2, 0], [-7, 1, -4])","explanation_zh":"結構基礎核對=([-6, 3, -4], [-1, -2, 0], [-7, 1, -4])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224007,"parameters":{"left":[-6,3,-4],"right":[-1,-2,0],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-6,3,-4],\"right\":[-1,-2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0008","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[-5,-3,-5],\"right\":[0,-1,1]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([-5, -3, -5], [0, -1, 1], [-5, -4, -4])","explanation_zh":"結構基礎核對=([-5, -3, -5], [0, -1, 1], [-5, -4, -4])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224008,"parameters":{"left":[-5,-3,-5],"right":[0,-1,1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-5,-3,-5],\"right\":[0,-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0009","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[-4,-2,-6],\"right\":[1,0,2]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([-4, -2, -6], [1, 0, 2], [-3, -2, -4])","explanation_zh":"結構基礎核對=([-4, -2, -6], [1, 0, 2], [-3, -2, -4])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224009,"parameters":{"left":[-4,-2,-6],"right":[1,0,2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-4,-2,-6],\"right\":[1,0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0010","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[-3,-1,-7],\"right\":[2,1,0]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([-3, -1, -7], [2, 1, 0], [-1, 0, -7])","explanation_zh":"結構基礎核對=([-3, -1, -7], [2, 1, 0], [-1, 0, -7])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224010,"parameters":{"left":[-3,-1,-7],"right":[2,1,0],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-3,-1,-7],\"right\":[2,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0011","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[-2,0,-8],\"right\":[-2,2,1]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([-2, 0, -8], [-2, 2, 1], [-4, 2, -7])","explanation_zh":"結構基礎核對=([-2, 0, -8], [-2, 2, 1], [-4, 2, -7])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224011,"parameters":{"left":[-2,0,-8],"right":[-2,2,1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-2,0,-8],\"right\":[-2,2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0012","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[-1,1,-9],\"right\":[-1,3,2]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([-1, 1, -9], [-1, 3, 2], [-2, 4, -7])","explanation_zh":"結構基礎核對=([-1, 1, -9], [-1, 3, 2], [-2, 4, -7])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224012,"parameters":{"left":[-1,1,-9],"right":[-1,3,2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-1,1,-9],\"right\":[-1,3,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0013","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[0,2,-10],\"right\":[0,4,0]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, -10], [0, 4, 0], [0, 6, -10])","explanation_zh":"結構基礎核對=([0, 2, -10], [0, 4, 0], [0, 6, -10])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224013,"parameters":{"left":[0,2,-10],"right":[0,4,0],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[0,2,-10],\"right\":[0,4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0014","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[1,3,-11],\"right\":[1,5,1]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 3, -11], [1, 5, 1], [2, 8, -10])","explanation_zh":"結構基礎核對=([1, 3, -11], [1, 5, 1], [2, 8, -10])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224014,"parameters":{"left":[1,3,-11],"right":[1,5,1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[1,3,-11],\"right\":[1,5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0015","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[2,-3,-12],\"right\":[2,6,2]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([2, -3, -12], [2, 6, 2], [4, 3, -10])","explanation_zh":"結構基礎核對=([2, -3, -12], [2, 6, 2], [4, 3, -10])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224015,"parameters":{"left":[2,-3,-12],"right":[2,6,2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[2,-3,-12],\"right\":[2,6,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0016","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[3,-2,-13],\"right\":[-2,7,0]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -13], [-2, 7, 0], [1, 5, -13])","explanation_zh":"結構基礎核對=([3, -2, -13], [-2, 7, 0], [1, 5, -13])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224016,"parameters":{"left":[3,-2,-13],"right":[-2,7,0],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[3,-2,-13],\"right\":[-2,7,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0017","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[4,-1,-14],\"right\":[-1,8,1]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -14], [-1, 8, 1], [3, 7, -13])","explanation_zh":"結構基礎核對=([4, -1, -14], [-1, 8, 1], [3, 7, -13])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224017,"parameters":{"left":[4,-1,-14],"right":[-1,8,1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[4,-1,-14],\"right\":[-1,8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0018","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[5,0,-15],\"right\":[0,9,2]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, -15], [0, 9, 2], [5, 9, -13])","explanation_zh":"結構基礎核對=([5, 0, -15], [0, 9, 2], [5, 9, -13])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224018,"parameters":{"left":[5,0,-15],"right":[0,9,2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[5,0,-15],\"right\":[0,9,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0019","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[6,1,-16],\"right\":[1,10,0]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, -16], [1, 10, 0], [7, 11, -16])","explanation_zh":"結構基礎核對=([6, 1, -16], [1, 10, 0], [7, 11, -16])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224019,"parameters":{"left":[6,1,-16],"right":[1,10,0],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[6,1,-16],\"right\":[1,10,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0020","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[7,2,-17],\"right\":[2,11,1]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, -17], [2, 11, 1], [9, 13, -16])","explanation_zh":"結構基礎核對=([7, 2, -17], [2, 11, 1], [9, 13, -16])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224020,"parameters":{"left":[7,2,-17],"right":[2,11,1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[7,2,-17],\"right\":[2,11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0021","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[8,3,-18],\"right\":[-2,12,2]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([8, 3, -18], [-2, 12, 2], [6, 15, -16])","explanation_zh":"結構基礎核對=([8, 3, -18], [-2, 12, 2], [6, 15, -16])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224021,"parameters":{"left":[8,3,-18],"right":[-2,12,2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[8,3,-18],\"right\":[-2,12,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0022","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[9,-3,-19],\"right\":[-1,13,0]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([9, -3, -19], [-1, 13, 0], [8, 10, -19])","explanation_zh":"結構基礎核對=([9, -3, -19], [-1, 13, 0], [8, 10, -19])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224022,"parameters":{"left":[9,-3,-19],"right":[-1,13,0],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[9,-3,-19],\"right\":[-1,13,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0023","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[10,-2,-20],\"right\":[0,14,1]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([10, -2, -20], [0, 14, 1], [10, 12, -19])","explanation_zh":"結構基礎核對=([10, -2, -20], [0, 14, 1], [10, 12, -19])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224023,"parameters":{"left":[10,-2,-20],"right":[0,14,1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[10,-2,-20],\"right\":[0,14,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0024","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[11,-1,-21],\"right\":[1,15,2]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([11, -1, -21], [1, 15, 2], [12, 14, -19])","explanation_zh":"結構基礎核對=([11, -1, -21], [1, 15, 2], [12, 14, -19])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224024,"parameters":{"left":[11,-1,-21],"right":[1,15,2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[11,-1,-21],\"right\":[1,15,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0025","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-addition；資料={\"left\":[12,0,-22],\"right\":[2,16,0]}。請依 [left,right,result] 欄序回答。","answer":{"kind":"expression","value":"([12, 0, -22], [2, 16, 0], [14, 16, -22])","explanation_zh":"結構基礎核對=([12, 0, -22], [2, 16, 0], [14, 16, -22])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-addition","seed":202610224025,"parameters":{"left":[12,0,-22],"right":[2,16,0],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-addition","semantic_template":"qf-str22-vectorspace-addition/{\"left\":[12,0,-22],\"right\":[2,16,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0026","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-3,\"vector\":[-12,-2,3]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(-3, [-12, -2, 3], [36, 6, -9])","explanation_zh":"結構基礎核對=(-3, [-12, -2, 3], [36, 6, -9])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224026,"parameters":{"scalar":-3,"vector":[-12,-2,3],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-3,\"vector\":[-12,-2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0027","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-2,\"vector\":[-11,-1,2]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(-2, [-11, -1, 2], [22, 2, -4])","explanation_zh":"結構基礎核對=(-2, [-11, -1, 2], [22, 2, -4])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224027,"parameters":{"scalar":-2,"vector":[-11,-1,2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-2,\"vector\":[-11,-1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0028","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-1,\"vector\":[-10,0,1]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(-1, [-10, 0, 1], [10, 0, -1])","explanation_zh":"結構基礎核對=(-1, [-10, 0, 1], [10, 0, -1])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224028,"parameters":{"scalar":-1,"vector":[-10,0,1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-1,\"vector\":[-10,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0029","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":0,\"vector\":[-9,1,0]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(0, [-9, 1, 0], [0, 0, 0])","explanation_zh":"結構基礎核對=(0, [-9, 1, 0], [0, 0, 0])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224029,"parameters":{"scalar":0,"vector":[-9,1,0],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":0,\"vector\":[-9,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0030","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":1,\"vector\":[-8,2,-1]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(1, [-8, 2, -1], [-8, 2, -1])","explanation_zh":"結構基礎核對=(1, [-8, 2, -1], [-8, 2, -1])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224030,"parameters":{"scalar":1,"vector":[-8,2,-1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":1,\"vector\":[-8,2,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0031","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":2,\"vector\":[-7,-2,-2]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(2, [-7, -2, -2], [-14, -4, -4])","explanation_zh":"結構基礎核對=(2, [-7, -2, -2], [-14, -4, -4])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224031,"parameters":{"scalar":2,"vector":[-7,-2,-2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":2,\"vector\":[-7,-2,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0032","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":3,\"vector\":[-6,-1,-3]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(3, [-6, -1, -3], [-18, -3, -9])","explanation_zh":"結構基礎核對=(3, [-6, -1, -3], [-18, -3, -9])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224032,"parameters":{"scalar":3,"vector":[-6,-1,-3],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":3,\"vector\":[-6,-1,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0033","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-3,\"vector\":[-5,0,-4]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(-3, [-5, 0, -4], [15, 0, 12])","explanation_zh":"結構基礎核對=(-3, [-5, 0, -4], [15, 0, 12])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224033,"parameters":{"scalar":-3,"vector":[-5,0,-4],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-3,\"vector\":[-5,0,-4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0034","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-2,\"vector\":[-4,1,-5]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(-2, [-4, 1, -5], [8, -2, 10])","explanation_zh":"結構基礎核對=(-2, [-4, 1, -5], [8, -2, 10])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224034,"parameters":{"scalar":-2,"vector":[-4,1,-5],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-2,\"vector\":[-4,1,-5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0035","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-1,\"vector\":[-3,2,-6]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(-1, [-3, 2, -6], [3, -2, 6])","explanation_zh":"結構基礎核對=(-1, [-3, 2, -6], [3, -2, 6])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224035,"parameters":{"scalar":-1,"vector":[-3,2,-6],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-1,\"vector\":[-3,2,-6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0036","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":0,\"vector\":[-2,-2,-7]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(0, [-2, -2, -7], [0, 0, 0])","explanation_zh":"結構基礎核對=(0, [-2, -2, -7], [0, 0, 0])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224036,"parameters":{"scalar":0,"vector":[-2,-2,-7],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":0,\"vector\":[-2,-2,-7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0037","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":1,\"vector\":[-1,-1,-8]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(1, [-1, -1, -8], [-1, -1, -8])","explanation_zh":"結構基礎核對=(1, [-1, -1, -8], [-1, -1, -8])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224037,"parameters":{"scalar":1,"vector":[-1,-1,-8],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":1,\"vector\":[-1,-1,-8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0038","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":2,\"vector\":[0,0,-9]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0, -9], [0, 0, -18])","explanation_zh":"結構基礎核對=(2, [0, 0, -9], [0, 0, -18])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224038,"parameters":{"scalar":2,"vector":[0,0,-9],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":2,\"vector\":[0,0,-9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0039","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":3,\"vector\":[1,1,-10]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(3, [1, 1, -10], [3, 3, -30])","explanation_zh":"結構基礎核對=(3, [1, 1, -10], [3, 3, -30])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224039,"parameters":{"scalar":3,"vector":[1,1,-10],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":3,\"vector\":[1,1,-10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0040","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-3,\"vector\":[2,2,-11]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(-3, [2, 2, -11], [-6, -6, 33])","explanation_zh":"結構基礎核對=(-3, [2, 2, -11], [-6, -6, 33])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224040,"parameters":{"scalar":-3,"vector":[2,2,-11],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-3,\"vector\":[2,2,-11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0041","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-2,\"vector\":[3,-2,-12]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(-2, [3, -2, -12], [-6, 4, 24])","explanation_zh":"結構基礎核對=(-2, [3, -2, -12], [-6, 4, 24])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224041,"parameters":{"scalar":-2,"vector":[3,-2,-12],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-2,\"vector\":[3,-2,-12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0042","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-1,\"vector\":[4,-1,-13]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(-1, [4, -1, -13], [-4, 1, 13])","explanation_zh":"結構基礎核對=(-1, [4, -1, -13], [-4, 1, 13])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224042,"parameters":{"scalar":-1,"vector":[4,-1,-13],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-1,\"vector\":[4,-1,-13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0043","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":0,\"vector\":[5,0,-14]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(0, [5, 0, -14], [0, 0, 0])","explanation_zh":"結構基礎核對=(0, [5, 0, -14], [0, 0, 0])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224043,"parameters":{"scalar":0,"vector":[5,0,-14],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":0,\"vector\":[5,0,-14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0044","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":1,\"vector\":[6,1,-15]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(1, [6, 1, -15], [6, 1, -15])","explanation_zh":"結構基礎核對=(1, [6, 1, -15], [6, 1, -15])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224044,"parameters":{"scalar":1,"vector":[6,1,-15],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":1,\"vector\":[6,1,-15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0045","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":2,\"vector\":[7,2,-16]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(2, [7, 2, -16], [14, 4, -32])","explanation_zh":"結構基礎核對=(2, [7, 2, -16], [14, 4, -32])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224045,"parameters":{"scalar":2,"vector":[7,2,-16],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":2,\"vector\":[7,2,-16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0046","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":3,\"vector\":[8,-2,-17]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(3, [8, -2, -17], [24, -6, -51])","explanation_zh":"結構基礎核對=(3, [8, -2, -17], [24, -6, -51])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224046,"parameters":{"scalar":3,"vector":[8,-2,-17],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":3,\"vector\":[8,-2,-17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0047","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-3,\"vector\":[9,-1,-18]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(-3, [9, -1, -18], [-27, 3, 54])","explanation_zh":"結構基礎核對=(-3, [9, -1, -18], [-27, 3, 54])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224047,"parameters":{"scalar":-3,"vector":[9,-1,-18],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-3,\"vector\":[9,-1,-18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0048","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-2,\"vector\":[10,0,-19]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(-2, [10, 0, -19], [-20, 0, 38])","explanation_zh":"結構基礎核對=(-2, [10, 0, -19], [-20, 0, 38])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224048,"parameters":{"scalar":-2,"vector":[10,0,-19],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-2,\"vector\":[10,0,-19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0049","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-1,\"vector\":[11,1,-20]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(-1, [11, 1, -20], [-11, -1, 20])","explanation_zh":"結構基礎核對=(-1, [11, 1, -20], [-11, -1, 20])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224049,"parameters":{"scalar":-1,"vector":[11,1,-20],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-1,\"vector\":[11,1,-20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0050","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":0,\"vector\":[12,2,-21]}。請依 [scalar,vector,result] 欄序回答。","answer":{"kind":"expression","value":"(0, [12, 2, -21], [0, 0, 0])","explanation_zh":"結構基礎核對=(0, [12, 2, -21], [0, 0, 0])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224050,"parameters":{"scalar":0,"vector":[12,2,-21],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-multiplication","semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":0,\"vector\":[12,2,-21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0051","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[-12,-2,3]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, 3], [12, 2, -3], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([-12, -2, 3], [12, 2, -3], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224051,"parameters":{"vector":[-12,-2,3],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-12,-2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0052","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[-11,-1,2]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, 2], [11, 1, -2], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([-11, -1, 2], [11, 1, -2], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224052,"parameters":{"vector":[-11,-1,2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-11,-1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0053","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[-10,0,1]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, 1], [10, 0, -1], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([-10, 0, 1], [10, 0, -1], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224053,"parameters":{"vector":[-10,0,1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-10,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0054","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[-9,1,0]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 0], [9, -1, 0], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([-9, 1, 0], [9, -1, 0], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224054,"parameters":{"vector":[-9,1,0],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-9,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0055","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[-8,2,-1]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, -1], [8, -2, 1], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([-8, 2, -1], [8, -2, 1], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224055,"parameters":{"vector":[-8,2,-1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-8,2,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0056","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[-7,-2,-2]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, -2], [7, 2, 2], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([-7, -2, -2], [7, 2, 2], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224056,"parameters":{"vector":[-7,-2,-2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-7,-2,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0057","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[-6,-1,-3]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, -3], [6, 1, 3], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([-6, -1, -3], [6, 1, 3], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224057,"parameters":{"vector":[-6,-1,-3],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-6,-1,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0058","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[-5,0,-4]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, -4], [5, 0, 4], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([-5, 0, -4], [5, 0, 4], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224058,"parameters":{"vector":[-5,0,-4],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-5,0,-4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0059","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[-4,1,-5]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, -5], [4, -1, 5], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([-4, 1, -5], [4, -1, 5], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224059,"parameters":{"vector":[-4,1,-5],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-4,1,-5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0060","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[-3,2,-6]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, -6], [3, -2, 6], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([-3, 2, -6], [3, -2, 6], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224060,"parameters":{"vector":[-3,2,-6],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-3,2,-6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0061","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[-2,-2,-7]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, -7], [2, 2, 7], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([-2, -2, -7], [2, 2, 7], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224061,"parameters":{"vector":[-2,-2,-7],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-2,-2,-7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0062","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[-1,-1,-8]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, -8], [1, 1, 8], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([-1, -1, -8], [1, 1, 8], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224062,"parameters":{"vector":[-1,-1,-8],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-1,-1,-8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0063","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[0,0,-9]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, -9], [0, 0, 9], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([0, 0, -9], [0, 0, 9], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224063,"parameters":{"vector":[0,0,-9],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[0,0,-9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0064","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[1,1,-10]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, -10], [-1, -1, 10], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([1, 1, -10], [-1, -1, 10], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224064,"parameters":{"vector":[1,1,-10],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[1,1,-10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0065","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[2,2,-11]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -11], [-2, -2, 11], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([2, 2, -11], [-2, -2, 11], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224065,"parameters":{"vector":[2,2,-11],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[2,2,-11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0066","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[3,-2,-12]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -12], [-3, 2, 12], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([3, -2, -12], [-3, 2, 12], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224066,"parameters":{"vector":[3,-2,-12],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[3,-2,-12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0067","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[4,-1,-13]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -13], [-4, 1, 13], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([4, -1, -13], [-4, 1, 13], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224067,"parameters":{"vector":[4,-1,-13],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[4,-1,-13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0068","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[5,0,-14]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, -14], [-5, 0, 14], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([5, 0, -14], [-5, 0, 14], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224068,"parameters":{"vector":[5,0,-14],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[5,0,-14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0069","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[6,1,-15]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, -15], [-6, -1, 15], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([6, 1, -15], [-6, -1, 15], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224069,"parameters":{"vector":[6,1,-15],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[6,1,-15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0070","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[7,2,-16]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, -16], [-7, -2, 16], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([7, 2, -16], [-7, -2, 16], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224070,"parameters":{"vector":[7,2,-16],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[7,2,-16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0071","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[8,-2,-17]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, -17], [-8, 2, 17], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([8, -2, -17], [-8, 2, 17], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224071,"parameters":{"vector":[8,-2,-17],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[8,-2,-17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0072","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[9,-1,-18]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, -18], [-9, 1, 18], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([9, -1, -18], [-9, 1, 18], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224072,"parameters":{"vector":[9,-1,-18],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[9,-1,-18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0073","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[10,0,-19]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, -19], [-10, 0, 19], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([10, 0, -19], [-10, 0, 19], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224073,"parameters":{"vector":[10,0,-19],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[10,0,-19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0074","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[11,1,-20]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, -20], [-11, -1, 20], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([11, 1, -20], [-11, -1, 20], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224074,"parameters":{"vector":[11,1,-20],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[11,1,-20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0075","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-additive-inverse；資料={\"vector\":[12,2,-21]}。請依 [vector,inverse,sum,zero] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, -21], [-12, -2, 21], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=([12, 2, -21], [-12, -2, 21], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224075,"parameters":{"vector":[12,2,-21],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-additive-inverse","semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[12,2,-21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0076","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":-3,\"b\":-2,\"vector\":[-12,-1,2]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(-3, -2, [-12, -1, 2], [60, 5, -10], [60, 5, -10], 是)","explanation_zh":"結構基礎核對=(-3, -2, [-12, -1, 2], [60, 5, -10], [60, 5, -10], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224076,"parameters":{"a":-3,"b":-2,"vector":[-12,-1,2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-3,\"b\":-2,\"vector\":[-12,-1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0077","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":-2,\"b\":-1,\"vector\":[-11,0,1]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, -1, [-11, 0, 1], [33, 0, -3], [33, 0, -3], 是)","explanation_zh":"結構基礎核對=(-2, -1, [-11, 0, 1], [33, 0, -3], [33, 0, -3], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224077,"parameters":{"a":-2,"b":-1,"vector":[-11,0,1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-2,\"b\":-1,\"vector\":[-11,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0078","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":-1,\"b\":0,\"vector\":[-10,1,0]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, 0, [-10, 1, 0], [10, -1, 0], [10, -1, 0], 是)","explanation_zh":"結構基礎核對=(-1, 0, [-10, 1, 0], [10, -1, 0], [10, -1, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224078,"parameters":{"a":-1,"b":0,"vector":[-10,1,0],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-1,\"b\":0,\"vector\":[-10,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0079","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":0,\"b\":1,\"vector\":[-9,2,-1]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [-9, 2, -1], [-9, 2, -1], [-9, 2, -1], 是)","explanation_zh":"結構基礎核對=(0, 1, [-9, 2, -1], [-9, 2, -1], [-9, 2, -1], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224079,"parameters":{"a":0,"b":1,"vector":[-9,2,-1],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":0,\"b\":1,\"vector\":[-9,2,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0080","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":1,\"b\":2,\"vector\":[-8,-1,-2]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [-8, -1, -2], [-24, -3, -6], [-24, -3, -6], 是)","explanation_zh":"結構基礎核對=(1, 2, [-8, -1, -2], [-24, -3, -6], [-24, -3, -6], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224080,"parameters":{"a":1,"b":2,"vector":[-8,-1,-2],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":1,\"b\":2,\"vector\":[-8,-1,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0081","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":2,\"b\":-2,\"vector\":[-7,0,-3]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, -2, [-7, 0, -3], [0, 0, 0], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=(2, -2, [-7, 0, -3], [0, 0, 0], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224081,"parameters":{"a":2,"b":-2,"vector":[-7,0,-3],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":2,\"b\":-2,\"vector\":[-7,0,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0082","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":3,\"b\":-1,\"vector\":[-6,1,-4]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(3, -1, [-6, 1, -4], [-12, 2, -8], [-12, 2, -8], 是)","explanation_zh":"結構基礎核對=(3, -1, [-6, 1, -4], [-12, 2, -8], [-12, 2, -8], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224082,"parameters":{"a":3,"b":-1,"vector":[-6,1,-4],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":3,\"b\":-1,\"vector\":[-6,1,-4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0083","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":-3,\"b\":0,\"vector\":[-5,2,-5]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(-3, 0, [-5, 2, -5], [15, -6, 15], [15, -6, 15], 是)","explanation_zh":"結構基礎核對=(-3, 0, [-5, 2, -5], [15, -6, 15], [15, -6, 15], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224083,"parameters":{"a":-3,"b":0,"vector":[-5,2,-5],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-3,\"b\":0,\"vector\":[-5,2,-5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0084","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":-2,\"b\":1,\"vector\":[-4,-1,-6]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, 1, [-4, -1, -6], [4, 1, 6], [4, 1, 6], 是)","explanation_zh":"結構基礎核對=(-2, 1, [-4, -1, -6], [4, 1, 6], [4, 1, 6], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224084,"parameters":{"a":-2,"b":1,"vector":[-4,-1,-6],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-2,\"b\":1,\"vector\":[-4,-1,-6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0085","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":-1,\"b\":2,\"vector\":[-3,0,-7]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, 2, [-3, 0, -7], [-3, 0, -7], [-3, 0, -7], 是)","explanation_zh":"結構基礎核對=(-1, 2, [-3, 0, -7], [-3, 0, -7], [-3, 0, -7], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224085,"parameters":{"a":-1,"b":2,"vector":[-3,0,-7],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-1,\"b\":2,\"vector\":[-3,0,-7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0086","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":0,\"b\":-2,\"vector\":[-2,1,-8]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, -2, [-2, 1, -8], [4, -2, 16], [4, -2, 16], 是)","explanation_zh":"結構基礎核對=(0, -2, [-2, 1, -8], [4, -2, 16], [4, -2, 16], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224086,"parameters":{"a":0,"b":-2,"vector":[-2,1,-8],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":0,\"b\":-2,\"vector\":[-2,1,-8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0087","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":1,\"b\":-1,\"vector\":[-1,2,-9]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, -1, [-1, 2, -9], [0, 0, 0], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=(1, -1, [-1, 2, -9], [0, 0, 0], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224087,"parameters":{"a":1,"b":-1,"vector":[-1,2,-9],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":1,\"b\":-1,\"vector\":[-1,2,-9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0088","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":2,\"b\":0,\"vector\":[0,-1,-10]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [0, -1, -10], [0, -2, -20], [0, -2, -20], 是)","explanation_zh":"結構基礎核對=(2, 0, [0, -1, -10], [0, -2, -20], [0, -2, -20], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224088,"parameters":{"a":2,"b":0,"vector":[0,-1,-10],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":2,\"b\":0,\"vector\":[0,-1,-10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0089","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":3,\"b\":1,\"vector\":[1,0,-11]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(3, 1, [1, 0, -11], [4, 0, -44], [4, 0, -44], 是)","explanation_zh":"結構基礎核對=(3, 1, [1, 0, -11], [4, 0, -44], [4, 0, -44], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224089,"parameters":{"a":3,"b":1,"vector":[1,0,-11],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":3,\"b\":1,\"vector\":[1,0,-11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0090","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":-3,\"b\":2,\"vector\":[2,1,-12]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(-3, 2, [2, 1, -12], [-2, -1, 12], [-2, -1, 12], 是)","explanation_zh":"結構基礎核對=(-3, 2, [2, 1, -12], [-2, -1, 12], [-2, -1, 12], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224090,"parameters":{"a":-3,"b":2,"vector":[2,1,-12],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-3,\"b\":2,\"vector\":[2,1,-12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0091","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":-2,\"b\":-2,\"vector\":[3,2,-13]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, -2, [3, 2, -13], [-12, -8, 52], [-12, -8, 52], 是)","explanation_zh":"結構基礎核對=(-2, -2, [3, 2, -13], [-12, -8, 52], [-12, -8, 52], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224091,"parameters":{"a":-2,"b":-2,"vector":[3,2,-13],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-2,\"b\":-2,\"vector\":[3,2,-13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0092","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":-1,\"b\":-1,\"vector\":[4,-1,-14]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, -1, [4, -1, -14], [-8, 2, 28], [-8, 2, 28], 是)","explanation_zh":"結構基礎核對=(-1, -1, [4, -1, -14], [-8, 2, 28], [-8, 2, 28], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224092,"parameters":{"a":-1,"b":-1,"vector":[4,-1,-14],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-1,\"b\":-1,\"vector\":[4,-1,-14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0093","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":0,\"b\":0,\"vector\":[5,0,-15]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, [5, 0, -15], [0, 0, 0], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=(0, 0, [5, 0, -15], [0, 0, 0], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224093,"parameters":{"a":0,"b":0,"vector":[5,0,-15],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":0,\"b\":0,\"vector\":[5,0,-15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0094","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":1,\"b\":1,\"vector\":[6,1,-16]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [6, 1, -16], [12, 2, -32], [12, 2, -32], 是)","explanation_zh":"結構基礎核對=(1, 1, [6, 1, -16], [12, 2, -32], [12, 2, -32], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224094,"parameters":{"a":1,"b":1,"vector":[6,1,-16],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":1,\"b\":1,\"vector\":[6,1,-16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0095","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":2,\"b\":2,\"vector\":[7,2,-17]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [7, 2, -17], [28, 8, -68], [28, 8, -68], 是)","explanation_zh":"結構基礎核對=(2, 2, [7, 2, -17], [28, 8, -68], [28, 8, -68], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224095,"parameters":{"a":2,"b":2,"vector":[7,2,-17],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":2,\"b\":2,\"vector\":[7,2,-17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0096","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":3,\"b\":-2,\"vector\":[8,-1,-18]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(3, -2, [8, -1, -18], [8, -1, -18], [8, -1, -18], 是)","explanation_zh":"結構基礎核對=(3, -2, [8, -1, -18], [8, -1, -18], [8, -1, -18], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224096,"parameters":{"a":3,"b":-2,"vector":[8,-1,-18],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":3,\"b\":-2,\"vector\":[8,-1,-18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0097","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":-3,\"b\":-1,\"vector\":[9,0,-19]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(-3, -1, [9, 0, -19], [-36, 0, 76], [-36, 0, 76], 是)","explanation_zh":"結構基礎核對=(-3, -1, [9, 0, -19], [-36, 0, 76], [-36, 0, 76], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224097,"parameters":{"a":-3,"b":-1,"vector":[9,0,-19],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-3,\"b\":-1,\"vector\":[9,0,-19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0098","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":-2,\"b\":0,\"vector\":[10,1,-20]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, 0, [10, 1, -20], [-20, -2, 40], [-20, -2, 40], 是)","explanation_zh":"結構基礎核對=(-2, 0, [10, 1, -20], [-20, -2, 40], [-20, -2, 40], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224098,"parameters":{"a":-2,"b":0,"vector":[10,1,-20],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-2,\"b\":0,\"vector\":[10,1,-20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0099","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":-1,\"b\":1,\"vector\":[11,2,-21]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, 1, [11, 2, -21], [0, 0, 0], [0, 0, 0], 是)","explanation_zh":"結構基礎核對=(-1, 1, [11, 2, -21], [0, 0, 0], [0, 0, 0], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224099,"parameters":{"a":-1,"b":1,"vector":[11,2,-21],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-1,\"b\":1,\"vector\":[11,2,-21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261022-r04-real-vector-space-0100","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-vector-space"],"title_zh":"實座標向量空間","stem_zh":"固定維度實座標空間。操作=vectorspace-scalar-distributivity；資料={\"a\":0,\"b\":2,\"vector\":[12,-1,-22]}。請依 [a,b,vector,left,right,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 2, [12, -1, -22], [24, -2, -44], [24, -2, -44], 是)","explanation_zh":"結構基礎核對=(0, 2, [12, -1, -22], [24, -2, -44], [24, -2, -44], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實座標向量空間"]},"variant":{"family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224100,"parameters":{"a":0,"b":2,"vector":[12,-1,-22],"domain":"Finite_coordinate_real_vector_space","operation":"vectorspace-scalar-distributivity","semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":0,\"b\":2,\"vector\":[12,-1,-22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}