{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0001","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[1,-1],\"vector\":[2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, -1], [2, 3], -1, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([1, -1], [2, 3], -1, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052001,"parameters":{"coefficients":[1,-1],"vector":[2,3],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[1,-1],\"vector\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0002","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[2,-1],\"vector\":[2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, -1], [2, 4], 0, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([2, -1], [2, 4], 0, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052002,"parameters":{"coefficients":[2,-1],"vector":[2,4],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[2,-1],\"vector\":[2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0003","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[3,-1],\"vector\":[2,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, -1], [2, 5], 1, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([3, -1], [2, 5], 1, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052003,"parameters":{"coefficients":[3,-1],"vector":[2,5],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[3,-1],\"vector\":[2,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0004","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[4,-1],\"vector\":[2,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, -1], [2, 6], 2, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([4, -1], [2, 6], 2, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052004,"parameters":{"coefficients":[4,-1],"vector":[2,6],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[4,-1],\"vector\":[2,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0005","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[5,-1],\"vector\":[2,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, -1], [2, 7], 3, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([5, -1], [2, 7], 3, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052005,"parameters":{"coefficients":[5,-1],"vector":[2,7],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[5,-1],\"vector\":[2,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0006","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[6,-1],\"vector\":[2,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, -1], [2, 8], 4, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([6, -1], [2, 8], 4, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052006,"parameters":{"coefficients":[6,-1],"vector":[2,8],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[6,-1],\"vector\":[2,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0007","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[7,-1],\"vector\":[2,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, -1], [2, 9], 5, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([7, -1], [2, 9], 5, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052007,"parameters":{"coefficients":[7,-1],"vector":[2,9],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[7,-1],\"vector\":[2,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0008","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[8,-1],\"vector\":[2,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, -1], [2, 10], 6, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([8, -1], [2, 10], 6, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052008,"parameters":{"coefficients":[8,-1],"vector":[2,10],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[8,-1],\"vector\":[2,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0009","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[9,-1],\"vector\":[2,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, -1], [2, 11], 7, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([9, -1], [2, 11], 7, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052009,"parameters":{"coefficients":[9,-1],"vector":[2,11],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[9,-1],\"vector\":[2,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0010","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[10,-1],\"vector\":[2,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, -1], [2, 12], 8, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([10, -1], [2, 12], 8, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052010,"parameters":{"coefficients":[10,-1],"vector":[2,12],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[10,-1],\"vector\":[2,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0011","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[11,-1],\"vector\":[2,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11, -1], [2, 13], 9, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([11, -1], [2, 13], 9, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052011,"parameters":{"coefficients":[11,-1],"vector":[2,13],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[11,-1],\"vector\":[2,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0012","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[12,-1],\"vector\":[2,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, -1], [2, 14], 10, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([12, -1], [2, 14], 10, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052012,"parameters":{"coefficients":[12,-1],"vector":[2,14],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[12,-1],\"vector\":[2,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0013","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[13,-1],\"vector\":[2,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([13, -1], [2, 15], 11, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([13, -1], [2, 15], 11, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052013,"parameters":{"coefficients":[13,-1],"vector":[2,15],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[13,-1],\"vector\":[2,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0014","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[14,-1],\"vector\":[2,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([14, -1], [2, 16], 12, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([14, -1], [2, 16], 12, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052014,"parameters":{"coefficients":[14,-1],"vector":[2,16],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[14,-1],\"vector\":[2,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0015","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[15,-1],\"vector\":[2,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([15, -1], [2, 17], 13, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([15, -1], [2, 17], 13, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052015,"parameters":{"coefficients":[15,-1],"vector":[2,17],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[15,-1],\"vector\":[2,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0016","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[16,-1],\"vector\":[2,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([16, -1], [2, 18], 14, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([16, -1], [2, 18], 14, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052016,"parameters":{"coefficients":[16,-1],"vector":[2,18],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[16,-1],\"vector\":[2,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0017","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[17,-1],\"vector\":[2,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([17, -1], [2, 19], 15, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([17, -1], [2, 19], 15, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052017,"parameters":{"coefficients":[17,-1],"vector":[2,19],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[17,-1],\"vector\":[2,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0018","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[18,-1],\"vector\":[2,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([18, -1], [2, 20], 16, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([18, -1], [2, 20], 16, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052018,"parameters":{"coefficients":[18,-1],"vector":[2,20],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[18,-1],\"vector\":[2,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0019","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[19,-1],\"vector\":[2,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([19, -1], [2, 21], 17, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([19, -1], [2, 21], 17, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052019,"parameters":{"coefficients":[19,-1],"vector":[2,21],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[19,-1],\"vector\":[2,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0020","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[20,-1],\"vector\":[2,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([20, -1], [2, 22], 18, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([20, -1], [2, 22], 18, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052020,"parameters":{"coefficients":[20,-1],"vector":[2,22],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[20,-1],\"vector\":[2,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0021","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[21,-1],\"vector\":[2,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([21, -1], [2, 23], 19, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([21, -1], [2, 23], 19, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052021,"parameters":{"coefficients":[21,-1],"vector":[2,23],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[21,-1],\"vector\":[2,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0022","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[22,-1],\"vector\":[2,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([22, -1], [2, 24], 20, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([22, -1], [2, 24], 20, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052022,"parameters":{"coefficients":[22,-1],"vector":[2,24],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[22,-1],\"vector\":[2,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0023","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[23,-1],\"vector\":[2,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([23, -1], [2, 25], 21, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([23, -1], [2, 25], 21, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052023,"parameters":{"coefficients":[23,-1],"vector":[2,25],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[23,-1],\"vector\":[2,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0024","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[24,-1],\"vector\":[2,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([24, -1], [2, 26], 22, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([24, -1], [2, 26], 22, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052024,"parameters":{"coefficients":[24,-1],"vector":[2,26],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[24,-1],\"vector\":[2,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0025","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[25,-1],\"vector\":[2,27]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([25, -1], [2, 27], 23, linear_functional_not_primal_vector)","explanation_zh":"Wave5第五層核對=([25, -1], [2, 27], 23, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-functional-evaluation","seed":202611052025,"parameters":{"coefficients":[25,-1],"vector":[2,27],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-functional-evaluation","semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[25,-1],\"vector\":[2,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0026","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[1,1]],\"candidate_dual\":[[1,-1],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052026,"parameters":{"basis":[[1,0],[1,1]],"candidate_dual":[[1,-1],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[1,1]],\"candidate_dual\":[[1,-1],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0027","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[2,1]],\"candidate_dual\":[[1,-2],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052027,"parameters":{"basis":[[1,0],[2,1]],"candidate_dual":[[1,-2],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[2,1]],\"candidate_dual\":[[1,-2],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0028","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[3,1]],\"candidate_dual\":[[1,-3],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052028,"parameters":{"basis":[[1,0],[3,1]],"candidate_dual":[[1,-3],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[3,1]],\"candidate_dual\":[[1,-3],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0029","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[4,1]],\"candidate_dual\":[[1,-4],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052029,"parameters":{"basis":[[1,0],[4,1]],"candidate_dual":[[1,-4],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[4,1]],\"candidate_dual\":[[1,-4],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0030","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[5,1]],\"candidate_dual\":[[1,-5],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052030,"parameters":{"basis":[[1,0],[5,1]],"candidate_dual":[[1,-5],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[5,1]],\"candidate_dual\":[[1,-5],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0031","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[6,1]],\"candidate_dual\":[[1,-6],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052031,"parameters":{"basis":[[1,0],[6,1]],"candidate_dual":[[1,-6],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[6,1]],\"candidate_dual\":[[1,-6],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0032","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[7,1]],\"candidate_dual\":[[1,-7],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052032,"parameters":{"basis":[[1,0],[7,1]],"candidate_dual":[[1,-7],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[7,1]],\"candidate_dual\":[[1,-7],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0033","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[8,1]],\"candidate_dual\":[[1,-8],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052033,"parameters":{"basis":[[1,0],[8,1]],"candidate_dual":[[1,-8],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[8,1]],\"candidate_dual\":[[1,-8],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0034","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[9,1]],\"candidate_dual\":[[1,-9],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052034,"parameters":{"basis":[[1,0],[9,1]],"candidate_dual":[[1,-9],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[9,1]],\"candidate_dual\":[[1,-9],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0035","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[10,1]],\"candidate_dual\":[[1,-10],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052035,"parameters":{"basis":[[1,0],[10,1]],"candidate_dual":[[1,-10],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[10,1]],\"candidate_dual\":[[1,-10],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0036","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[11,1]],\"candidate_dual\":[[1,-11],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052036,"parameters":{"basis":[[1,0],[11,1]],"candidate_dual":[[1,-11],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[11,1]],\"candidate_dual\":[[1,-11],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0037","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[12,1]],\"candidate_dual\":[[1,-12],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052037,"parameters":{"basis":[[1,0],[12,1]],"candidate_dual":[[1,-12],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[12,1]],\"candidate_dual\":[[1,-12],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0038","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[13,1]],\"candidate_dual\":[[1,-13],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052038,"parameters":{"basis":[[1,0],[13,1]],"candidate_dual":[[1,-13],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[13,1]],\"candidate_dual\":[[1,-13],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0039","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[14,1]],\"candidate_dual\":[[1,-14],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052039,"parameters":{"basis":[[1,0],[14,1]],"candidate_dual":[[1,-14],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[14,1]],\"candidate_dual\":[[1,-14],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0040","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[15,1]],\"candidate_dual\":[[1,-15],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052040,"parameters":{"basis":[[1,0],[15,1]],"candidate_dual":[[1,-15],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[15,1]],\"candidate_dual\":[[1,-15],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0041","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[16,1]],\"candidate_dual\":[[1,-16],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052041,"parameters":{"basis":[[1,0],[16,1]],"candidate_dual":[[1,-16],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[16,1]],\"candidate_dual\":[[1,-16],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0042","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[17,1]],\"candidate_dual\":[[1,-17],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052042,"parameters":{"basis":[[1,0],[17,1]],"candidate_dual":[[1,-17],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[17,1]],\"candidate_dual\":[[1,-17],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0043","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[18,1]],\"candidate_dual\":[[1,-18],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052043,"parameters":{"basis":[[1,0],[18,1]],"candidate_dual":[[1,-18],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[18,1]],\"candidate_dual\":[[1,-18],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0044","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[19,1]],\"candidate_dual\":[[1,-19],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052044,"parameters":{"basis":[[1,0],[19,1]],"candidate_dual":[[1,-19],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[19,1]],\"candidate_dual\":[[1,-19],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0045","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[20,1]],\"candidate_dual\":[[1,-20],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052045,"parameters":{"basis":[[1,0],[20,1]],"candidate_dual":[[1,-20],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[20,1]],\"candidate_dual\":[[1,-20],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0046","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[21,1]],\"candidate_dual\":[[1,-21],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052046,"parameters":{"basis":[[1,0],[21,1]],"candidate_dual":[[1,-21],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[21,1]],\"candidate_dual\":[[1,-21],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0047","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[22,1]],\"candidate_dual\":[[1,-22],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052047,"parameters":{"basis":[[1,0],[22,1]],"candidate_dual":[[1,-22],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[22,1]],\"candidate_dual\":[[1,-22],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0048","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[23,1]],\"candidate_dual\":[[1,-23],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052048,"parameters":{"basis":[[1,0],[23,1]],"candidate_dual":[[1,-23],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[23,1]],\"candidate_dual\":[[1,-23],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0049","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[24,1]],\"candidate_dual\":[[1,-24],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052049,"parameters":{"basis":[[1,0],[24,1]],"candidate_dual":[[1,-24],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[24,1]],\"candidate_dual\":[[1,-24],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0050","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-dual-basis-certificate；資料={\"basis\":[[1,0],[25,1]],\"candidate_dual\":[[1,-25],[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)","explanation_zh":"Wave5第五層核對=([[1, 0], [0, 1]], 是, phi_i_v_j_equals_Kronecker_delta)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052050,"parameters":{"basis":[[1,0],[25,1]],"candidate_dual":[[1,-25],[0,1]],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-dual-basis-certificate","semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[25,1]],\"candidate_dual\":[[1,-25],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0051","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[1,-2],\"u\":[1,2],\"v\":[3,-1],\"alpha\":1,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-13, -13, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-13, -13, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052051,"parameters":{"coefficients":[1,-2],"u":[1,2],"v":[3,-1],"alpha":1,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":1,\"beta\":-2,\"coefficients\":[1,-2],\"u\":[1,2],\"v\":[3,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0052","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[2,-2],\"u\":[1,3],\"v\":[4,-1],\"alpha\":2,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-28, -28, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-28, -28, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052052,"parameters":{"coefficients":[2,-2],"u":[1,3],"v":[4,-1],"alpha":2,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":2,\"beta\":-2,\"coefficients\":[2,-2],\"u\":[1,3],\"v\":[4,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0053","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[3,-2],\"u\":[1,4],\"v\":[5,-1],\"alpha\":3,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-49, -49, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-49, -49, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052053,"parameters":{"coefficients":[3,-2],"u":[1,4],"v":[5,-1],"alpha":3,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":3,\"beta\":-2,\"coefficients\":[3,-2],\"u\":[1,4],\"v\":[5,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0054","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[4,-2],\"u\":[1,5],\"v\":[6,-1],\"alpha\":4,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-76, -76, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-76, -76, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052054,"parameters":{"coefficients":[4,-2],"u":[1,5],"v":[6,-1],"alpha":4,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":4,\"beta\":-2,\"coefficients\":[4,-2],\"u\":[1,5],\"v\":[6,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0055","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[5,-2],\"u\":[1,6],\"v\":[7,-1],\"alpha\":5,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-109, -109, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-109, -109, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052055,"parameters":{"coefficients":[5,-2],"u":[1,6],"v":[7,-1],"alpha":5,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":5,\"beta\":-2,\"coefficients\":[5,-2],\"u\":[1,6],\"v\":[7,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0056","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[6,-2],\"u\":[1,7],\"v\":[8,-1],\"alpha\":6,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-148, -148, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-148, -148, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052056,"parameters":{"coefficients":[6,-2],"u":[1,7],"v":[8,-1],"alpha":6,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":6,\"beta\":-2,\"coefficients\":[6,-2],\"u\":[1,7],\"v\":[8,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0057","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[7,-2],\"u\":[1,8],\"v\":[9,-1],\"alpha\":7,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-193, -193, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-193, -193, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052057,"parameters":{"coefficients":[7,-2],"u":[1,8],"v":[9,-1],"alpha":7,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":7,\"beta\":-2,\"coefficients\":[7,-2],\"u\":[1,8],\"v\":[9,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0058","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[8,-2],\"u\":[1,9],\"v\":[10,-1],\"alpha\":8,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-244, -244, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-244, -244, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052058,"parameters":{"coefficients":[8,-2],"u":[1,9],"v":[10,-1],"alpha":8,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":8,\"beta\":-2,\"coefficients\":[8,-2],\"u\":[1,9],\"v\":[10,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0059","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[9,-2],\"u\":[1,10],\"v\":[11,-1],\"alpha\":9,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-301, -301, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-301, -301, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052059,"parameters":{"coefficients":[9,-2],"u":[1,10],"v":[11,-1],"alpha":9,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":9,\"beta\":-2,\"coefficients\":[9,-2],\"u\":[1,10],\"v\":[11,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0060","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[10,-2],\"u\":[1,11],\"v\":[12,-1],\"alpha\":10,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-364, -364, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-364, -364, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052060,"parameters":{"coefficients":[10,-2],"u":[1,11],"v":[12,-1],"alpha":10,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":10,\"beta\":-2,\"coefficients\":[10,-2],\"u\":[1,11],\"v\":[12,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0061","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[11,-2],\"u\":[1,12],\"v\":[13,-1],\"alpha\":11,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-433, -433, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-433, -433, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052061,"parameters":{"coefficients":[11,-2],"u":[1,12],"v":[13,-1],"alpha":11,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":11,\"beta\":-2,\"coefficients\":[11,-2],\"u\":[1,12],\"v\":[13,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0062","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[12,-2],\"u\":[1,13],\"v\":[14,-1],\"alpha\":12,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-508, -508, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-508, -508, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052062,"parameters":{"coefficients":[12,-2],"u":[1,13],"v":[14,-1],"alpha":12,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":12,\"beta\":-2,\"coefficients\":[12,-2],\"u\":[1,13],\"v\":[14,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0063","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[13,-2],\"u\":[1,14],\"v\":[15,-1],\"alpha\":13,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-589, -589, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-589, -589, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052063,"parameters":{"coefficients":[13,-2],"u":[1,14],"v":[15,-1],"alpha":13,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":13,\"beta\":-2,\"coefficients\":[13,-2],\"u\":[1,14],\"v\":[15,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0064","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[14,-2],\"u\":[1,15],\"v\":[16,-1],\"alpha\":14,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-676, -676, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-676, -676, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052064,"parameters":{"coefficients":[14,-2],"u":[1,15],"v":[16,-1],"alpha":14,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":14,\"beta\":-2,\"coefficients\":[14,-2],\"u\":[1,15],\"v\":[16,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0065","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[15,-2],\"u\":[1,16],\"v\":[17,-1],\"alpha\":15,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-769, -769, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-769, -769, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052065,"parameters":{"coefficients":[15,-2],"u":[1,16],"v":[17,-1],"alpha":15,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":15,\"beta\":-2,\"coefficients\":[15,-2],\"u\":[1,16],\"v\":[17,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0066","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[16,-2],\"u\":[1,17],\"v\":[18,-1],\"alpha\":16,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-868, -868, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-868, -868, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052066,"parameters":{"coefficients":[16,-2],"u":[1,17],"v":[18,-1],"alpha":16,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":16,\"beta\":-2,\"coefficients\":[16,-2],\"u\":[1,17],\"v\":[18,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0067","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[17,-2],\"u\":[1,18],\"v\":[19,-1],\"alpha\":17,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-973, -973, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-973, -973, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052067,"parameters":{"coefficients":[17,-2],"u":[1,18],"v":[19,-1],"alpha":17,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":17,\"beta\":-2,\"coefficients\":[17,-2],\"u\":[1,18],\"v\":[19,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0068","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[18,-2],\"u\":[1,19],\"v\":[20,-1],\"alpha\":18,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1084, -1084, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-1084, -1084, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052068,"parameters":{"coefficients":[18,-2],"u":[1,19],"v":[20,-1],"alpha":18,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":18,\"beta\":-2,\"coefficients\":[18,-2],\"u\":[1,19],\"v\":[20,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0069","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[19,-2],\"u\":[1,20],\"v\":[21,-1],\"alpha\":19,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1201, -1201, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-1201, -1201, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052069,"parameters":{"coefficients":[19,-2],"u":[1,20],"v":[21,-1],"alpha":19,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":19,\"beta\":-2,\"coefficients\":[19,-2],\"u\":[1,20],\"v\":[21,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0070","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[20,-2],\"u\":[1,21],\"v\":[22,-1],\"alpha\":20,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1324, -1324, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-1324, -1324, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052070,"parameters":{"coefficients":[20,-2],"u":[1,21],"v":[22,-1],"alpha":20,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":20,\"beta\":-2,\"coefficients\":[20,-2],\"u\":[1,21],\"v\":[22,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0071","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[21,-2],\"u\":[1,22],\"v\":[23,-1],\"alpha\":21,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1453, -1453, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-1453, -1453, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052071,"parameters":{"coefficients":[21,-2],"u":[1,22],"v":[23,-1],"alpha":21,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":21,\"beta\":-2,\"coefficients\":[21,-2],\"u\":[1,22],\"v\":[23,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0072","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[22,-2],\"u\":[1,23],\"v\":[24,-1],\"alpha\":22,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1588, -1588, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-1588, -1588, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052072,"parameters":{"coefficients":[22,-2],"u":[1,23],"v":[24,-1],"alpha":22,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":22,\"beta\":-2,\"coefficients\":[22,-2],\"u\":[1,23],\"v\":[24,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0073","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[23,-2],\"u\":[1,24],\"v\":[25,-1],\"alpha\":23,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1729, -1729, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-1729, -1729, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052073,"parameters":{"coefficients":[23,-2],"u":[1,24],"v":[25,-1],"alpha":23,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":23,\"beta\":-2,\"coefficients\":[23,-2],\"u\":[1,24],\"v\":[25,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0074","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[24,-2],\"u\":[1,25],\"v\":[26,-1],\"alpha\":24,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1876, -1876, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-1876, -1876, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052074,"parameters":{"coefficients":[24,-2],"u":[1,25],"v":[26,-1],"alpha":24,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":24,\"beta\":-2,\"coefficients\":[24,-2],\"u\":[1,25],\"v\":[26,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0075","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[25,-2],\"u\":[1,26],\"v\":[27,-1],\"alpha\":25,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2029, -2029, 是, row_covector_gives_universal_linearity_certificate)","explanation_zh":"Wave5第五層核對=(-2029, -2029, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-linearity-certificate","seed":202611052075,"parameters":{"coefficients":[25,-2],"u":[1,26],"v":[27,-1],"alpha":25,"beta":-2,"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-linearity-certificate","semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":25,\"beta\":-2,\"coefficients\":[25,-2],\"u\":[1,26],\"v\":[27,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0076","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,1],[0,1]],\"primal_coordinates\":[2,3],\"covector_coordinates\":[2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-1, 3], [2, 3], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-1, 3], [2, 3], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052076,"parameters":{"change_of_basis":[[1,1],[0,1]],"primal_coordinates":[2,3],"covector_coordinates":[2,1],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,1],[0,1]],\"covector_coordinates\":[2,1],\"primal_coordinates\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0077","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,2],[0,1]],\"primal_coordinates\":[3,3],\"covector_coordinates\":[2,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-3, 3], [2, 6], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-3, 3], [2, 6], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052077,"parameters":{"change_of_basis":[[1,2],[0,1]],"primal_coordinates":[3,3],"covector_coordinates":[2,2],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,2],[0,1]],\"covector_coordinates\":[2,2],\"primal_coordinates\":[3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0078","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,3],[0,1]],\"primal_coordinates\":[4,3],\"covector_coordinates\":[2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-5, 3], [2, 9], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-5, 3], [2, 9], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052078,"parameters":{"change_of_basis":[[1,3],[0,1]],"primal_coordinates":[4,3],"covector_coordinates":[2,3],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,3],[0,1]],\"covector_coordinates\":[2,3],\"primal_coordinates\":[4,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0079","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,4],[0,1]],\"primal_coordinates\":[5,3],\"covector_coordinates\":[2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-7, 3], [2, 12], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-7, 3], [2, 12], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052079,"parameters":{"change_of_basis":[[1,4],[0,1]],"primal_coordinates":[5,3],"covector_coordinates":[2,4],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,4],[0,1]],\"covector_coordinates\":[2,4],\"primal_coordinates\":[5,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0080","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,5],[0,1]],\"primal_coordinates\":[6,3],\"covector_coordinates\":[2,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-9, 3], [2, 15], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-9, 3], [2, 15], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052080,"parameters":{"change_of_basis":[[1,5],[0,1]],"primal_coordinates":[6,3],"covector_coordinates":[2,5],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,5],[0,1]],\"covector_coordinates\":[2,5],\"primal_coordinates\":[6,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0081","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,6],[0,1]],\"primal_coordinates\":[7,3],\"covector_coordinates\":[2,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-11, 3], [2, 18], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-11, 3], [2, 18], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052081,"parameters":{"change_of_basis":[[1,6],[0,1]],"primal_coordinates":[7,3],"covector_coordinates":[2,6],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,6],[0,1]],\"covector_coordinates\":[2,6],\"primal_coordinates\":[7,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0082","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,7],[0,1]],\"primal_coordinates\":[8,3],\"covector_coordinates\":[2,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-13, 3], [2, 21], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-13, 3], [2, 21], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052082,"parameters":{"change_of_basis":[[1,7],[0,1]],"primal_coordinates":[8,3],"covector_coordinates":[2,7],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,7],[0,1]],\"covector_coordinates\":[2,7],\"primal_coordinates\":[8,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0083","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,8],[0,1]],\"primal_coordinates\":[9,3],\"covector_coordinates\":[2,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-15, 3], [2, 24], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-15, 3], [2, 24], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052083,"parameters":{"change_of_basis":[[1,8],[0,1]],"primal_coordinates":[9,3],"covector_coordinates":[2,8],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,8],[0,1]],\"covector_coordinates\":[2,8],\"primal_coordinates\":[9,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0084","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,9],[0,1]],\"primal_coordinates\":[10,3],\"covector_coordinates\":[2,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-17, 3], [2, 27], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-17, 3], [2, 27], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052084,"parameters":{"change_of_basis":[[1,9],[0,1]],"primal_coordinates":[10,3],"covector_coordinates":[2,9],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,9],[0,1]],\"covector_coordinates\":[2,9],\"primal_coordinates\":[10,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0085","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,10],[0,1]],\"primal_coordinates\":[11,3],\"covector_coordinates\":[2,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-19, 3], [2, 30], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-19, 3], [2, 30], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052085,"parameters":{"change_of_basis":[[1,10],[0,1]],"primal_coordinates":[11,3],"covector_coordinates":[2,10],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,10],[0,1]],\"covector_coordinates\":[2,10],\"primal_coordinates\":[11,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0086","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,11],[0,1]],\"primal_coordinates\":[12,3],\"covector_coordinates\":[2,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-21, 3], [2, 33], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-21, 3], [2, 33], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052086,"parameters":{"change_of_basis":[[1,11],[0,1]],"primal_coordinates":[12,3],"covector_coordinates":[2,11],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,11],[0,1]],\"covector_coordinates\":[2,11],\"primal_coordinates\":[12,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0087","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,12],[0,1]],\"primal_coordinates\":[13,3],\"covector_coordinates\":[2,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-23, 3], [2, 36], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-23, 3], [2, 36], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052087,"parameters":{"change_of_basis":[[1,12],[0,1]],"primal_coordinates":[13,3],"covector_coordinates":[2,12],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,12],[0,1]],\"covector_coordinates\":[2,12],\"primal_coordinates\":[13,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0088","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,13],[0,1]],\"primal_coordinates\":[14,3],\"covector_coordinates\":[2,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-25, 3], [2, 39], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-25, 3], [2, 39], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052088,"parameters":{"change_of_basis":[[1,13],[0,1]],"primal_coordinates":[14,3],"covector_coordinates":[2,13],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,13],[0,1]],\"covector_coordinates\":[2,13],\"primal_coordinates\":[14,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0089","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,14],[0,1]],\"primal_coordinates\":[15,3],\"covector_coordinates\":[2,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-27, 3], [2, 42], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-27, 3], [2, 42], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052089,"parameters":{"change_of_basis":[[1,14],[0,1]],"primal_coordinates":[15,3],"covector_coordinates":[2,14],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,14],[0,1]],\"covector_coordinates\":[2,14],\"primal_coordinates\":[15,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0090","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,15],[0,1]],\"primal_coordinates\":[16,3],\"covector_coordinates\":[2,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-29, 3], [2, 45], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-29, 3], [2, 45], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052090,"parameters":{"change_of_basis":[[1,15],[0,1]],"primal_coordinates":[16,3],"covector_coordinates":[2,15],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,15],[0,1]],\"covector_coordinates\":[2,15],\"primal_coordinates\":[16,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0091","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,16],[0,1]],\"primal_coordinates\":[17,3],\"covector_coordinates\":[2,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-31, 3], [2, 48], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-31, 3], [2, 48], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052091,"parameters":{"change_of_basis":[[1,16],[0,1]],"primal_coordinates":[17,3],"covector_coordinates":[2,16],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,16],[0,1]],\"covector_coordinates\":[2,16],\"primal_coordinates\":[17,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0092","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,17],[0,1]],\"primal_coordinates\":[18,3],\"covector_coordinates\":[2,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-33, 3], [2, 51], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-33, 3], [2, 51], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052092,"parameters":{"change_of_basis":[[1,17],[0,1]],"primal_coordinates":[18,3],"covector_coordinates":[2,17],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,17],[0,1]],\"covector_coordinates\":[2,17],\"primal_coordinates\":[18,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0093","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,18],[0,1]],\"primal_coordinates\":[19,3],\"covector_coordinates\":[2,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-35, 3], [2, 54], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-35, 3], [2, 54], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052093,"parameters":{"change_of_basis":[[1,18],[0,1]],"primal_coordinates":[19,3],"covector_coordinates":[2,18],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,18],[0,1]],\"covector_coordinates\":[2,18],\"primal_coordinates\":[19,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0094","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,19],[0,1]],\"primal_coordinates\":[20,3],\"covector_coordinates\":[2,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-37, 3], [2, 57], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-37, 3], [2, 57], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052094,"parameters":{"change_of_basis":[[1,19],[0,1]],"primal_coordinates":[20,3],"covector_coordinates":[2,19],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,19],[0,1]],\"covector_coordinates\":[2,19],\"primal_coordinates\":[20,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0095","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,20],[0,1]],\"primal_coordinates\":[21,3],\"covector_coordinates\":[2,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-39, 3], [2, 60], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-39, 3], [2, 60], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052095,"parameters":{"change_of_basis":[[1,20],[0,1]],"primal_coordinates":[21,3],"covector_coordinates":[2,20],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,20],[0,1]],\"covector_coordinates\":[2,20],\"primal_coordinates\":[21,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0096","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,21],[0,1]],\"primal_coordinates\":[22,3],\"covector_coordinates\":[2,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-41, 3], [2, 63], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-41, 3], [2, 63], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052096,"parameters":{"change_of_basis":[[1,21],[0,1]],"primal_coordinates":[22,3],"covector_coordinates":[2,21],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,21],[0,1]],\"covector_coordinates\":[2,21],\"primal_coordinates\":[22,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0097","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,22],[0,1]],\"primal_coordinates\":[23,3],\"covector_coordinates\":[2,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-43, 3], [2, 66], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-43, 3], [2, 66], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052097,"parameters":{"change_of_basis":[[1,22],[0,1]],"primal_coordinates":[23,3],"covector_coordinates":[2,22],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,22],[0,1]],\"covector_coordinates\":[2,22],\"primal_coordinates\":[23,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0098","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,23],[0,1]],\"primal_coordinates\":[24,3],\"covector_coordinates\":[2,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-45, 3], [2, 69], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-45, 3], [2, 69], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052098,"parameters":{"change_of_basis":[[1,23],[0,1]],"primal_coordinates":[24,3],"covector_coordinates":[2,23],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,23],[0,1]],\"covector_coordinates\":[2,23],\"primal_coordinates\":[24,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0099","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,24],[0,1]],\"primal_coordinates\":[25,3],\"covector_coordinates\":[2,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-47, 3], [2, 72], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-47, 3], [2, 72], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052099,"parameters":{"change_of_basis":[[1,24],[0,1]],"primal_coordinates":[25,3],"covector_coordinates":[2,24],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,24],[0,1]],\"covector_coordinates\":[2,24],\"primal_coordinates\":[25,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r02-dual-vector-space-0100","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","language":"zh-Hant","source_mko_ids":["mko-dual-vector-space"],"title_zh":"對偶向量空間","stem_zh":"有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,25],[0,1]],\"primal_coordinates\":[26,3],\"covector_coordinates\":[2,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-49, 3], [2, 75], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)","explanation_zh":"Wave5第五層核對=([-49, 3], [2, 75], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對偶向量空間"]},"variant":{"family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052100,"parameters":{"change_of_basis":[[1,25],[0,1]],"primal_coordinates":[26,3],"covector_coordinates":[2,25],"domain":"Finite_dimensional_real_dual_coordinates","operation":"dual-coordinate-change-boundary","semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,25],[0,1]],\"covector_coordinates\":[2,25],\"primal_coordinates\":[26,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}