{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0001","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(0, 0)，v=(0, 0)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"分量結果=(0, 0)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045001,"parameters":{"u":[0,0],"domain":"reals","operation":"vector_add","v":[0,0],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[0,0],\"v\":[0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0002","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-11, -1)，v=(11, -2)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, -3)","explanation_zh":"分量結果=(0, -3)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045002,"parameters":{"u":[-11,-1],"domain":"reals","operation":"vector_add","v":[11,-2],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-11,-1],\"v\":[11,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0003","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-10, 0)，v=(10, -1)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, -1)","explanation_zh":"分量結果=(0, -1)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045003,"parameters":{"u":[-10,0],"domain":"reals","operation":"vector_add","v":[10,-1],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-10,0],\"v\":[10,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0004","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-9, 1)，v=(9, 0)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 1)","explanation_zh":"分量結果=(0, 1)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045004,"parameters":{"u":[-9,1],"domain":"reals","operation":"vector_add","v":[9,0],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-9,1],\"v\":[9,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0005","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-8, 2)，v=(8, 1)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 3)","explanation_zh":"分量結果=(0, 3)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045005,"parameters":{"u":[-8,2],"domain":"reals","operation":"vector_add","v":[8,1],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-8,2],\"v\":[8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0006","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-7, -2)，v=(7, 2)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"分量結果=(0, 0)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045006,"parameters":{"u":[-7,-2],"domain":"reals","operation":"vector_add","v":[7,2],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-7,-2],\"v\":[7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0007","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-6, -1)，v=(6, 3)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 2)","explanation_zh":"分量結果=(0, 2)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045007,"parameters":{"u":[-6,-1],"domain":"reals","operation":"vector_add","v":[6,3],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-6,-1],\"v\":[6,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0008","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-5, 0)，v=(5, -3)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, -3)","explanation_zh":"分量結果=(0, -3)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045008,"parameters":{"u":[-5,0],"domain":"reals","operation":"vector_add","v":[5,-3],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-5,0],\"v\":[5,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0009","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-4, 1)，v=(4, -2)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, -1)","explanation_zh":"分量結果=(0, -1)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045009,"parameters":{"u":[-4,1],"domain":"reals","operation":"vector_add","v":[4,-2],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-4,1],\"v\":[4,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0010","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-3, 2)，v=(3, -1)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 1)","explanation_zh":"分量結果=(0, 1)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045010,"parameters":{"u":[-3,2],"domain":"reals","operation":"vector_add","v":[3,-1],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-3,2],\"v\":[3,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0011","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-2, -2)，v=(2, 0)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, -2)","explanation_zh":"分量結果=(0, -2)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045011,"parameters":{"u":[-2,-2],"domain":"reals","operation":"vector_add","v":[2,0],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-2,-2],\"v\":[2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0012","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-1, -1)，v=(1, 1)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"分量結果=(0, 0)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045012,"parameters":{"u":[-1,-1],"domain":"reals","operation":"vector_add","v":[1,1],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-1,-1],\"v\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0013","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(0, 0)，v=(0, 2)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 2)","explanation_zh":"分量結果=(0, 2)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045013,"parameters":{"u":[0,0],"domain":"reals","operation":"vector_add","v":[0,2],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[0,0],\"v\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0014","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(1, 1)，v=(-1, 3)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 4)","explanation_zh":"分量結果=(0, 4)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045014,"parameters":{"u":[1,1],"domain":"reals","operation":"vector_add","v":[-1,3],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-1,3],\"v\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0015","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(2, 2)，v=(-2, -3)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, -1)","explanation_zh":"分量結果=(0, -1)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045015,"parameters":{"u":[2,2],"domain":"reals","operation":"vector_add","v":[-2,-3],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-2,-3],\"v\":[2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0016","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(3, -2)，v=(-3, -2)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, -4)","explanation_zh":"分量結果=(0, -4)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045016,"parameters":{"u":[3,-2],"domain":"reals","operation":"vector_add","v":[-3,-2],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-3,-2],\"v\":[3,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0017","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(4, -1)，v=(-4, -1)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, -2)","explanation_zh":"分量結果=(0, -2)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045017,"parameters":{"u":[4,-1],"domain":"reals","operation":"vector_add","v":[-4,-1],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-4,-1],\"v\":[4,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0018","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(5, 0)，v=(-5, 0)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"分量結果=(0, 0)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045018,"parameters":{"u":[5,0],"domain":"reals","operation":"vector_add","v":[-5,0],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-5,0],\"v\":[5,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0019","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(6, 1)，v=(-6, 1)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 2)","explanation_zh":"分量結果=(0, 2)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045019,"parameters":{"u":[6,1],"domain":"reals","operation":"vector_add","v":[-6,1],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-6,1],\"v\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0020","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(7, 2)，v=(-7, 2)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 4)","explanation_zh":"分量結果=(0, 4)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045020,"parameters":{"u":[7,2],"domain":"reals","operation":"vector_add","v":[-7,2],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-7,2],\"v\":[7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0021","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(8, -2)，v=(-8, 3)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 1)","explanation_zh":"分量結果=(0, 1)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045021,"parameters":{"u":[8,-2],"domain":"reals","operation":"vector_add","v":[-8,3],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-8,3],\"v\":[8,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0022","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(9, -1)，v=(-9, -3)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, -4)","explanation_zh":"分量結果=(0, -4)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045022,"parameters":{"u":[9,-1],"domain":"reals","operation":"vector_add","v":[-9,-3],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-9,-3],\"v\":[9,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0023","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(10, 0)，v=(-10, -2)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, -2)","explanation_zh":"分量結果=(0, -2)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045023,"parameters":{"u":[10,0],"domain":"reals","operation":"vector_add","v":[-10,-2],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-10,-2],\"v\":[10,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0024","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(11, 1)，v=(-11, -1)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"分量結果=(0, 0)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045024,"parameters":{"u":[11,1],"domain":"reals","operation":"vector_add","v":[-11,-1],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-11,-1],\"v\":[11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0025","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的逐分量加法","stem_zh":"在固定基底下的實數平面向量 R²，u=(12, 2)，v=(-12, 0)。求 u+v 的精確分量。","answer":{"kind":"expression","value":"(0, 2)","explanation_zh":"分量結果=(0, 2)。逐分量做加法；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的逐分量加法"]},"variant":{"family_id":"qf-vector-coordinate-addition","seed":202610045025,"parameters":{"u":[12,2],"domain":"reals","operation":"vector_add","v":[-12,0],"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-12,0],\"v\":[12,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0026","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(0, 0)，純量 k=0。求 ku 的精確分量。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"ku=(0, 0)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045026,"parameters":{"u":[0,0],"domain":"reals","operation":"scalar_multiply","k":0,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":0,\"operation\":\"scalar_multiply\",\"u\":[0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0027","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-11, -1)，純量 k=-2。求 ku 的精確分量。","answer":{"kind":"expression","value":"(22, 2)","explanation_zh":"ku=(22, 2)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045027,"parameters":{"u":[-11,-1],"domain":"reals","operation":"scalar_multiply","k":-2,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-2,\"operation\":\"scalar_multiply\",\"u\":[-11,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0028","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-10, 0)，純量 k=-1。求 ku 的精確分量。","answer":{"kind":"expression","value":"(10, 0)","explanation_zh":"ku=(10, 0)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045028,"parameters":{"u":[-10,0],"domain":"reals","operation":"scalar_multiply","k":-1,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-1,\"operation\":\"scalar_multiply\",\"u\":[-10,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0029","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-9, 1)，純量 k=0。求 ku 的精確分量。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"ku=(0, 0)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045029,"parameters":{"u":[-9,1],"domain":"reals","operation":"scalar_multiply","k":0,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":0,\"operation\":\"scalar_multiply\",\"u\":[-9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0030","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-8, 2)，純量 k=1。求 ku 的精確分量。","answer":{"kind":"expression","value":"(-8, 2)","explanation_zh":"ku=(-8, 2)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045030,"parameters":{"u":[-8,2],"domain":"reals","operation":"scalar_multiply","k":1,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":1,\"operation\":\"scalar_multiply\",\"u\":[-8,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0031","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-7, -2)，純量 k=2。求 ku 的精確分量。","answer":{"kind":"expression","value":"(-14, -4)","explanation_zh":"ku=(-14, -4)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045031,"parameters":{"u":[-7,-2],"domain":"reals","operation":"scalar_multiply","k":2,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":2,\"operation\":\"scalar_multiply\",\"u\":[-7,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0032","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-6, -1)，純量 k=3。求 ku 的精確分量。","answer":{"kind":"expression","value":"(-18, -3)","explanation_zh":"ku=(-18, -3)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045032,"parameters":{"u":[-6,-1],"domain":"reals","operation":"scalar_multiply","k":3,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":3,\"operation\":\"scalar_multiply\",\"u\":[-6,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0033","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-5, 0)，純量 k=-3。求 ku 的精確分量。","answer":{"kind":"expression","value":"(15, 0)","explanation_zh":"ku=(15, 0)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045033,"parameters":{"u":[-5,0],"domain":"reals","operation":"scalar_multiply","k":-3,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-3,\"operation\":\"scalar_multiply\",\"u\":[-5,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0034","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-4, 1)，純量 k=-2。求 ku 的精確分量。","answer":{"kind":"expression","value":"(8, -2)","explanation_zh":"ku=(8, -2)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045034,"parameters":{"u":[-4,1],"domain":"reals","operation":"scalar_multiply","k":-2,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-2,\"operation\":\"scalar_multiply\",\"u\":[-4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0035","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-3, 2)，純量 k=-1。求 ku 的精確分量。","answer":{"kind":"expression","value":"(3, -2)","explanation_zh":"ku=(3, -2)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045035,"parameters":{"u":[-3,2],"domain":"reals","operation":"scalar_multiply","k":-1,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-1,\"operation\":\"scalar_multiply\",\"u\":[-3,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0036","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-2, -2)，純量 k=0。求 ku 的精確分量。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"ku=(0, 0)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045036,"parameters":{"u":[-2,-2],"domain":"reals","operation":"scalar_multiply","k":0,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":0,\"operation\":\"scalar_multiply\",\"u\":[-2,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0037","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-1, -1)，純量 k=1。求 ku 的精確分量。","answer":{"kind":"expression","value":"(-1, -1)","explanation_zh":"ku=(-1, -1)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045037,"parameters":{"u":[-1,-1],"domain":"reals","operation":"scalar_multiply","k":1,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":1,\"operation\":\"scalar_multiply\",\"u\":[-1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0038","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(0, 0)，純量 k=2。求 ku 的精確分量。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"ku=(0, 0)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045038,"parameters":{"u":[0,0],"domain":"reals","operation":"scalar_multiply","k":2,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":2,\"operation\":\"scalar_multiply\",\"u\":[0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0039","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(1, 1)，純量 k=3。求 ku 的精確分量。","answer":{"kind":"expression","value":"(3, 3)","explanation_zh":"ku=(3, 3)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045039,"parameters":{"u":[1,1],"domain":"reals","operation":"scalar_multiply","k":3,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":3,\"operation\":\"scalar_multiply\",\"u\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0040","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(2, 2)，純量 k=-3。求 ku 的精確分量。","answer":{"kind":"expression","value":"(-6, -6)","explanation_zh":"ku=(-6, -6)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045040,"parameters":{"u":[2,2],"domain":"reals","operation":"scalar_multiply","k":-3,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-3,\"operation\":\"scalar_multiply\",\"u\":[2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0041","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(3, -2)，純量 k=-2。求 ku 的精確分量。","answer":{"kind":"expression","value":"(-6, 4)","explanation_zh":"ku=(-6, 4)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045041,"parameters":{"u":[3,-2],"domain":"reals","operation":"scalar_multiply","k":-2,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-2,\"operation\":\"scalar_multiply\",\"u\":[3,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0042","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(4, -1)，純量 k=-1。求 ku 的精確分量。","answer":{"kind":"expression","value":"(-4, 1)","explanation_zh":"ku=(-4, 1)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045042,"parameters":{"u":[4,-1],"domain":"reals","operation":"scalar_multiply","k":-1,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-1,\"operation\":\"scalar_multiply\",\"u\":[4,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0043","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(5, 0)，純量 k=0。求 ku 的精確分量。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"ku=(0, 0)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045043,"parameters":{"u":[5,0],"domain":"reals","operation":"scalar_multiply","k":0,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":0,\"operation\":\"scalar_multiply\",\"u\":[5,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0044","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(6, 1)，純量 k=1。求 ku 的精確分量。","answer":{"kind":"expression","value":"(6, 1)","explanation_zh":"ku=(6, 1)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045044,"parameters":{"u":[6,1],"domain":"reals","operation":"scalar_multiply","k":1,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":1,\"operation\":\"scalar_multiply\",\"u\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0045","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(7, 2)，純量 k=2。求 ku 的精確分量。","answer":{"kind":"expression","value":"(14, 4)","explanation_zh":"ku=(14, 4)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045045,"parameters":{"u":[7,2],"domain":"reals","operation":"scalar_multiply","k":2,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":2,\"operation\":\"scalar_multiply\",\"u\":[7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0046","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(8, -2)，純量 k=3。求 ku 的精確分量。","answer":{"kind":"expression","value":"(24, -6)","explanation_zh":"ku=(24, -6)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045046,"parameters":{"u":[8,-2],"domain":"reals","operation":"scalar_multiply","k":3,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":3,\"operation\":\"scalar_multiply\",\"u\":[8,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0047","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(9, -1)，純量 k=-3。求 ku 的精確分量。","answer":{"kind":"expression","value":"(-27, 3)","explanation_zh":"ku=(-27, 3)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045047,"parameters":{"u":[9,-1],"domain":"reals","operation":"scalar_multiply","k":-3,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-3,\"operation\":\"scalar_multiply\",\"u\":[9,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0048","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(10, 0)，純量 k=-2。求 ku 的精確分量。","answer":{"kind":"expression","value":"(-20, 0)","explanation_zh":"ku=(-20, 0)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045048,"parameters":{"u":[10,0],"domain":"reals","operation":"scalar_multiply","k":-2,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-2,\"operation\":\"scalar_multiply\",\"u\":[10,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0049","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(11, 1)，純量 k=-1。求 ku 的精確分量。","answer":{"kind":"expression","value":"(-11, -1)","explanation_zh":"ku=(-11, -1)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045049,"parameters":{"u":[11,1],"domain":"reals","operation":"scalar_multiply","k":-1,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-1,\"operation\":\"scalar_multiply\",\"u\":[11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0050","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的純量乘法","stem_zh":"在固定基底下的實數平面向量 R²，u=(12, 2)，純量 k=0。求 ku 的精確分量。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"ku=(0, 0)。每個分量各乘 k，零純量給零向量；純量乘法不是內積，也不推論長度。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的純量乘法"]},"variant":{"family_id":"qf-vector-scalar-multiplication","seed":202610045050,"parameters":{"u":[12,2],"domain":"reals","operation":"scalar_multiply","k":0,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":0,\"operation\":\"scalar_multiply\",\"u\":[12,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0051","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-12, -2)，v=(-3, 12)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-9, -14)","explanation_zh":"分量結果=(-9, -14)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045051,"parameters":{"u":[-12,-2],"domain":"reals","operation":"vector_subtract","v":[-3,12],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-12,-2],\"v\":[-3,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0052","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-11, -1)，v=(-2, 11)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-9, -12)","explanation_zh":"分量結果=(-9, -12)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045052,"parameters":{"u":[-11,-1],"domain":"reals","operation":"vector_subtract","v":[-2,11],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-11,-1],\"v\":[-2,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0053","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-10, 0)，v=(-1, 10)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-9, -10)","explanation_zh":"分量結果=(-9, -10)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045053,"parameters":{"u":[-10,0],"domain":"reals","operation":"vector_subtract","v":[-1,10],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-10,0],\"v\":[-1,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0054","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-9, 1)，v=(0, 9)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-9, -8)","explanation_zh":"分量結果=(-9, -8)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045054,"parameters":{"u":[-9,1],"domain":"reals","operation":"vector_subtract","v":[0,9],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-9,1],\"v\":[0,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0055","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-8, 2)，v=(1, 8)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-9, -6)","explanation_zh":"分量結果=(-9, -6)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045055,"parameters":{"u":[-8,2],"domain":"reals","operation":"vector_subtract","v":[1,8],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-8,2],\"v\":[1,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0056","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-7, -2)，v=(2, 7)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-9, -9)","explanation_zh":"分量結果=(-9, -9)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045056,"parameters":{"u":[-7,-2],"domain":"reals","operation":"vector_subtract","v":[2,7],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-7,-2],\"v\":[2,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0057","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-6, -1)，v=(3, 6)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-9, -7)","explanation_zh":"分量結果=(-9, -7)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045057,"parameters":{"u":[-6,-1],"domain":"reals","operation":"vector_subtract","v":[3,6],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-6,-1],\"v\":[3,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0058","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-5, 0)，v=(-3, 5)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-2, -5)","explanation_zh":"分量結果=(-2, -5)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045058,"parameters":{"u":[-5,0],"domain":"reals","operation":"vector_subtract","v":[-3,5],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-5,0],\"v\":[-3,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0059","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-4, 1)，v=(-2, 4)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-2, -3)","explanation_zh":"分量結果=(-2, -3)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045059,"parameters":{"u":[-4,1],"domain":"reals","operation":"vector_subtract","v":[-2,4],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-4,1],\"v\":[-2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0060","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-3, 2)，v=(-1, 3)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-2, -1)","explanation_zh":"分量結果=(-2, -1)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045060,"parameters":{"u":[-3,2],"domain":"reals","operation":"vector_subtract","v":[-1,3],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-3,2],\"v\":[-1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0061","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-2, -2)，v=(0, 2)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-2, -4)","explanation_zh":"分量結果=(-2, -4)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045061,"parameters":{"u":[-2,-2],"domain":"reals","operation":"vector_subtract","v":[0,2],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-2,-2],\"v\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0062","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(-1, -1)，v=(1, 1)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-2, -2)","explanation_zh":"分量結果=(-2, -2)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045062,"parameters":{"u":[-1,-1],"domain":"reals","operation":"vector_subtract","v":[1,1],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-1,-1],\"v\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0063","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(0, 0)，v=(2, 0)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-2, 0)","explanation_zh":"分量結果=(-2, 0)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045063,"parameters":{"u":[0,0],"domain":"reals","operation":"vector_subtract","v":[2,0],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[0,0],\"v\":[2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0064","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(1, 1)，v=(3, -1)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(-2, 2)","explanation_zh":"分量結果=(-2, 2)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045064,"parameters":{"u":[1,1],"domain":"reals","operation":"vector_subtract","v":[3,-1],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[1,1],\"v\":[3,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0065","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(2, 2)，v=(-3, -2)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(5, 4)","explanation_zh":"分量結果=(5, 4)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045065,"parameters":{"u":[2,2],"domain":"reals","operation":"vector_subtract","v":[-3,-2],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[2,2],\"v\":[-3,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0066","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(3, -2)，v=(-2, -3)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(5, 1)","explanation_zh":"分量結果=(5, 1)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045066,"parameters":{"u":[3,-2],"domain":"reals","operation":"vector_subtract","v":[-2,-3],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[3,-2],\"v\":[-2,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0067","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(4, -1)，v=(-1, -4)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(5, 3)","explanation_zh":"分量結果=(5, 3)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045067,"parameters":{"u":[4,-1],"domain":"reals","operation":"vector_subtract","v":[-1,-4],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[4,-1],\"v\":[-1,-4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0068","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(5, 0)，v=(0, -5)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(5, 5)","explanation_zh":"分量結果=(5, 5)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045068,"parameters":{"u":[5,0],"domain":"reals","operation":"vector_subtract","v":[0,-5],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[5,0],\"v\":[0,-5]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0069","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(6, 1)，v=(1, -6)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(5, 7)","explanation_zh":"分量結果=(5, 7)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045069,"parameters":{"u":[6,1],"domain":"reals","operation":"vector_subtract","v":[1,-6],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[6,1],\"v\":[1,-6]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0070","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(7, 2)，v=(2, -7)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(5, 9)","explanation_zh":"分量結果=(5, 9)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045070,"parameters":{"u":[7,2],"domain":"reals","operation":"vector_subtract","v":[2,-7],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[7,2],\"v\":[2,-7]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0071","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(8, -2)，v=(3, -8)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(5, 6)","explanation_zh":"分量結果=(5, 6)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045071,"parameters":{"u":[8,-2],"domain":"reals","operation":"vector_subtract","v":[3,-8],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[8,-2],\"v\":[3,-8]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0072","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(9, -1)，v=(-3, -9)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(12, 8)","explanation_zh":"分量結果=(12, 8)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045072,"parameters":{"u":[9,-1],"domain":"reals","operation":"vector_subtract","v":[-3,-9],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[9,-1],\"v\":[-3,-9]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0073","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(10, 0)，v=(-2, -10)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(12, 10)","explanation_zh":"分量結果=(12, 10)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045073,"parameters":{"u":[10,0],"domain":"reals","operation":"vector_subtract","v":[-2,-10],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[10,0],\"v\":[-2,-10]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0074","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(11, 1)，v=(-1, -11)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(12, 12)","explanation_zh":"分量結果=(12, 12)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045074,"parameters":{"u":[11,1],"domain":"reals","operation":"vector_subtract","v":[-1,-11],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[11,1],\"v\":[-1,-11]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0075","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的有方向減法","stem_zh":"在固定基底下的實數平面向量 R²，u=(12, 2)，v=(0, -12)。求 u−v（不可交換順序） 的精確分量。","answer":{"kind":"expression","value":"(12, 14)","explanation_zh":"分量結果=(12, 14)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定基底向量的有方向減法"]},"variant":{"family_id":"qf-vector-coordinate-difference","seed":202610045075,"parameters":{"u":[12,2],"domain":"reals","operation":"vector_subtract","v":[0,-12],"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[12,2],\"v\":[0,-12]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0076","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(0, 0)，v=(0, 0)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"分量結果=是。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045076,"parameters":{"u":[0,0],"domain":"reals","operation":"vector_equality","v":[0,0],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[0,0],\"v\":[0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0077","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(-11, -1)，v=(-10, -2)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"分量結果=否。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045077,"parameters":{"u":[-11,-1],"domain":"reals","operation":"vector_equality","v":[-10,-2],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-10,-2],\"v\":[-11,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0078","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(-10, 0)，v=(-10, 0)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"分量結果=是。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045078,"parameters":{"u":[-10,0],"domain":"reals","operation":"vector_equality","v":[-10,0],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-10,0],\"v\":[-10,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0079","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(-9, 1)，v=(1, -9)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"分量結果=否。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045079,"parameters":{"u":[-9,1],"domain":"reals","operation":"vector_equality","v":[1,-9],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-9,1],\"v\":[1,-9]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0080","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(-8, 2)，v=(-8, 2)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"分量結果=是。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045080,"parameters":{"u":[-8,2],"domain":"reals","operation":"vector_equality","v":[-8,2],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-8,2],\"v\":[-8,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0081","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(-7, -2)，v=(-6, -3)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"分量結果=否。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045081,"parameters":{"u":[-7,-2],"domain":"reals","operation":"vector_equality","v":[-6,-3],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-6,-3],\"v\":[-7,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0082","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(-6, -1)，v=(-6, -1)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"分量結果=是。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045082,"parameters":{"u":[-6,-1],"domain":"reals","operation":"vector_equality","v":[-6,-1],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-6,-1],\"v\":[-6,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0083","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(-5, 0)，v=(-4, -1)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"分量結果=否。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045083,"parameters":{"u":[-5,0],"domain":"reals","operation":"vector_equality","v":[-4,-1],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-4,-1],\"v\":[-5,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0084","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(-4, 1)，v=(-4, 1)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"分量結果=是。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045084,"parameters":{"u":[-4,1],"domain":"reals","operation":"vector_equality","v":[-4,1],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-4,1],\"v\":[-4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0085","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(-3, 2)，v=(-2, 1)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"分量結果=否。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045085,"parameters":{"u":[-3,2],"domain":"reals","operation":"vector_equality","v":[-2,1],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-2,1],\"v\":[-3,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0086","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(-2, -2)，v=(-2, -2)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"分量結果=是。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045086,"parameters":{"u":[-2,-2],"domain":"reals","operation":"vector_equality","v":[-2,-2],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-2,-2],\"v\":[-2,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0087","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(-1, -1)，v=(0, -2)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"分量結果=否。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045087,"parameters":{"u":[-1,-1],"domain":"reals","operation":"vector_equality","v":[0,-2],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-1,-1],\"v\":[0,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0088","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(0, 4)，v=(0, 4)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"分量結果=是。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045088,"parameters":{"u":[0,4],"domain":"reals","operation":"vector_equality","v":[0,4],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[0,4],\"v\":[0,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r05-vector-0089","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-vector"],"title_zh":"固定基底向量的分量相等","stem_zh":"在固定基底下的實數平面向量 R²，u=(1, 1)，v=(2, 0)。u 與 v 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"分量結果=否。須兩個對應位置的分量都相等，分量和相等或任意交換不夠。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定基底向量的分量相等"]},"variant":{"family_id":"qf-vector-coordinate-equality","seed":202610045089,"parameters":{"u":[1,1],"domain":"reals","operation":"vector_equality","v":[2,0],"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[1,1],\"v\":[2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
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