{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0001","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[2,-1],[-1,2]],\"x\":[-2,-3],\"y\":[-5,12]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[2, -1], [-1, 2]], [-2, -3], [-5, 12], -43)","explanation_zh":"Wave4第二層核對=([[2, -1], [-1, 2]], [-2, -3], [-5, 12], -43)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284001,"parameters":{"matrix":[[2,-1],[-1,2]],"x":[-2,-3],"y":[-5,12],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[2,-1],[-1,2]],\"x\":[-2,-3],\"y\":[-5,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0002","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[3,0],[0,1]],\"x\":[-1,-2],\"y\":[-4,11]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[3, 0], [0, 1]], [-1, -2], [-4, 11], -10)","explanation_zh":"Wave4第二層核對=([[3, 0], [0, 1]], [-1, -2], [-4, 11], -10)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284002,"parameters":{"matrix":[[3,0],[0,1]],"x":[-1,-2],"y":[-4,11],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[3,0],[0,1]],\"x\":[-1,-2],\"y\":[-4,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0003","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[4,1],[1,2]],\"x\":[0,-1],\"y\":[-3,10]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[4, 1], [1, 2]], [0, -1], [-3, 10], -17)","explanation_zh":"Wave4第二層核對=([[4, 1], [1, 2]], [0, -1], [-3, 10], -17)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284003,"parameters":{"matrix":[[4,1],[1,2]],"x":[0,-1],"y":[-3,10],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[4,1],[1,2]],\"x\":[0,-1],\"y\":[-3,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0004","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[5,2],[2,5]],\"x\":[1,0],\"y\":[-2,9]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[5, 2], [2, 5]], [1, 0], [-2, 9], 8)","explanation_zh":"Wave4第二層核對=([[5, 2], [2, 5]], [1, 0], [-2, 9], 8)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284004,"parameters":{"matrix":[[5,2],[2,5]],"x":[1,0],"y":[-2,9],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[5,2],[2,5]],\"x\":[1,0],\"y\":[-2,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0005","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[6,-1],[-1,2]],\"x\":[2,1],\"y\":[-1,8]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[6, -1], [-1, 2]], [2, 1], [-1, 8], -11)","explanation_zh":"Wave4第二層核對=([[6, -1], [-1, 2]], [2, 1], [-1, 8], -11)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284005,"parameters":{"matrix":[[6,-1],[-1,2]],"x":[2,1],"y":[-1,8],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[6,-1],[-1,2]],\"x\":[2,1],\"y\":[-1,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0006","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[7,0],[0,1]],\"x\":[-2,2],\"y\":[0,7]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[7, 0], [0, 1]], [-2, 2], [0, 7], 14)","explanation_zh":"Wave4第二層核對=([[7, 0], [0, 1]], [-2, 2], [0, 7], 14)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284006,"parameters":{"matrix":[[7,0],[0,1]],"x":[-2,2],"y":[0,7],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[7,0],[0,1]],\"x\":[-2,2],\"y\":[0,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0007","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[8,1],[1,2]],\"x\":[-1,3],\"y\":[1,6]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[8, 1], [1, 2]], [-1, 3], [1, 6], 25)","explanation_zh":"Wave4第二層核對=([[8, 1], [1, 2]], [-1, 3], [1, 6], 25)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284007,"parameters":{"matrix":[[8,1],[1,2]],"x":[-1,3],"y":[1,6],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[8,1],[1,2]],\"x\":[-1,3],\"y\":[1,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0008","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[9,2],[2,5]],\"x\":[0,-3],\"y\":[2,5]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[9, 2], [2, 5]], [0, -3], [2, 5], -87)","explanation_zh":"Wave4第二層核對=([[9, 2], [2, 5]], [0, -3], [2, 5], -87)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284008,"parameters":{"matrix":[[9,2],[2,5]],"x":[0,-3],"y":[2,5],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[9,2],[2,5]],\"x\":[0,-3],\"y\":[2,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0009","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[10,-1],[-1,2]],\"x\":[1,-2],\"y\":[3,4]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[10, -1], [-1, 2]], [1, -2], [3, 4], 16)","explanation_zh":"Wave4第二層核對=([[10, -1], [-1, 2]], [1, -2], [3, 4], 16)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284009,"parameters":{"matrix":[[10,-1],[-1,2]],"x":[1,-2],"y":[3,4],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[10,-1],[-1,2]],\"x\":[1,-2],\"y\":[3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0010","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[11,0],[0,1]],\"x\":[2,-1],\"y\":[4,3]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[11, 0], [0, 1]], [2, -1], [4, 3], 85)","explanation_zh":"Wave4第二層核對=([[11, 0], [0, 1]], [2, -1], [4, 3], 85)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284010,"parameters":{"matrix":[[11,0],[0,1]],"x":[2,-1],"y":[4,3],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[11,0],[0,1]],\"x\":[2,-1],\"y\":[4,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0011","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[12,1],[1,2]],\"x\":[-2,0],\"y\":[5,2]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[12, 1], [1, 2]], [-2, 0], [5, 2], -124)","explanation_zh":"Wave4第二層核對=([[12, 1], [1, 2]], [-2, 0], [5, 2], -124)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284011,"parameters":{"matrix":[[12,1],[1,2]],"x":[-2,0],"y":[5,2],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[12,1],[1,2]],\"x\":[-2,0],\"y\":[5,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0012","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[13,2],[2,5]],\"x\":[-1,1],\"y\":[-5,1]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[13, 2], [2, 5]], [-1, 1], [-5, 1], 58)","explanation_zh":"Wave4第二層核對=([[13, 2], [2, 5]], [-1, 1], [-5, 1], 58)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284012,"parameters":{"matrix":[[13,2],[2,5]],"x":[-1,1],"y":[-5,1],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[13,2],[2,5]],\"x\":[-1,1],\"y\":[-5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0013","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[14,-1],[-1,2]],\"x\":[0,2],\"y\":[-4,0]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[14, -1], [-1, 2]], [0, 2], [-4, 0], 8)","explanation_zh":"Wave4第二層核對=([[14, -1], [-1, 2]], [0, 2], [-4, 0], 8)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284013,"parameters":{"matrix":[[14,-1],[-1,2]],"x":[0,2],"y":[-4,0],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[14,-1],[-1,2]],\"x\":[0,2],\"y\":[-4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0014","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[15,0],[0,1]],\"x\":[1,3],\"y\":[-3,-1]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[15, 0], [0, 1]], [1, 3], [-3, -1], -48)","explanation_zh":"Wave4第二層核對=([[15, 0], [0, 1]], [1, 3], [-3, -1], -48)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284014,"parameters":{"matrix":[[15,0],[0,1]],"x":[1,3],"y":[-3,-1],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[15,0],[0,1]],\"x\":[1,3],\"y\":[-3,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0015","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[16,1],[1,2]],\"x\":[2,-3],\"y\":[-2,-2]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[16, 1], [1, 2]], [2, -3], [-2, -2], -50)","explanation_zh":"Wave4第二層核對=([[16, 1], [1, 2]], [2, -3], [-2, -2], -50)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284015,"parameters":{"matrix":[[16,1],[1,2]],"x":[2,-3],"y":[-2,-2],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[16,1],[1,2]],\"x\":[2,-3],\"y\":[-2,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0016","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[17,2],[2,5]],\"x\":[-2,-2],\"y\":[-1,-3]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[17, 2], [2, 5]], [-2, -2], [-1, -3], 80)","explanation_zh":"Wave4第二層核對=([[17, 2], [2, 5]], [-2, -2], [-1, -3], 80)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284016,"parameters":{"matrix":[[17,2],[2,5]],"x":[-2,-2],"y":[-1,-3],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[17,2],[2,5]],\"x\":[-2,-2],\"y\":[-1,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0017","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[18,-1],[-1,2]],\"x\":[-1,-1],\"y\":[0,-4]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[18, -1], [-1, 2]], [-1, -1], [0, -4], 4)","explanation_zh":"Wave4第二層核對=([[18, -1], [-1, 2]], [-1, -1], [0, -4], 4)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284017,"parameters":{"matrix":[[18,-1],[-1,2]],"x":[-1,-1],"y":[0,-4],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[18,-1],[-1,2]],\"x\":[-1,-1],\"y\":[0,-4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0018","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[19,0],[0,1]],\"x\":[0,0],\"y\":[1,-5]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[19, 0], [0, 1]], [0, 0], [1, -5], 0)","explanation_zh":"Wave4第二層核對=([[19, 0], [0, 1]], [0, 0], [1, -5], 0)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284018,"parameters":{"matrix":[[19,0],[0,1]],"x":[0,0],"y":[1,-5],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[19,0],[0,1]],\"x\":[0,0],\"y\":[1,-5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0019","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[20,1],[1,2]],\"x\":[1,1],\"y\":[2,-6]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[20, 1], [1, 2]], [1, 1], [2, -6], 24)","explanation_zh":"Wave4第二層核對=([[20, 1], [1, 2]], [1, 1], [2, -6], 24)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284019,"parameters":{"matrix":[[20,1],[1,2]],"x":[1,1],"y":[2,-6],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[20,1],[1,2]],\"x\":[1,1],\"y\":[2,-6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0020","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[21,2],[2,5]],\"x\":[2,2],\"y\":[3,-7]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[21, 2], [2, 5]], [2, 2], [3, -7], 40)","explanation_zh":"Wave4第二層核對=([[21, 2], [2, 5]], [2, 2], [3, -7], 40)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284020,"parameters":{"matrix":[[21,2],[2,5]],"x":[2,2],"y":[3,-7],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[21,2],[2,5]],\"x\":[2,2],\"y\":[3,-7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0021","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[22,-1],[-1,2]],\"x\":[-2,3],\"y\":[4,-8]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[22, -1], [-1, 2]], [-2, 3], [4, -8], -252)","explanation_zh":"Wave4第二層核對=([[22, -1], [-1, 2]], [-2, 3], [4, -8], -252)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284021,"parameters":{"matrix":[[22,-1],[-1,2]],"x":[-2,3],"y":[4,-8],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[22,-1],[-1,2]],\"x\":[-2,3],\"y\":[4,-8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0022","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[23,0],[0,1]],\"x\":[-1,-3],\"y\":[5,-9]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[23, 0], [0, 1]], [-1, -3], [5, -9], -88)","explanation_zh":"Wave4第二層核對=([[23, 0], [0, 1]], [-1, -3], [5, -9], -88)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284022,"parameters":{"matrix":[[23,0],[0,1]],"x":[-1,-3],"y":[5,-9],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[23,0],[0,1]],\"x\":[-1,-3],\"y\":[5,-9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0023","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[24,1],[1,2]],\"x\":[0,-2],\"y\":[-5,-10]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[24, 1], [1, 2]], [0, -2], [-5, -10], 50)","explanation_zh":"Wave4第二層核對=([[24, 1], [1, 2]], [0, -2], [-5, -10], 50)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284023,"parameters":{"matrix":[[24,1],[1,2]],"x":[0,-2],"y":[-5,-10],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[24,1],[1,2]],\"x\":[0,-2],\"y\":[-5,-10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0024","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[25,2],[2,5]],\"x\":[1,-1],\"y\":[-4,-11]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[25, 2], [2, 5]], [1, -1], [-4, -11], -59)","explanation_zh":"Wave4第二層核對=([[25, 2], [2, 5]], [1, -1], [-4, -11], -59)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284024,"parameters":{"matrix":[[25,2],[2,5]],"x":[1,-1],"y":[-4,-11],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[25,2],[2,5]],\"x\":[1,-1],\"y\":[-4,-11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0025","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={\"matrix\":[[26,-1],[-1,2]],\"x\":[2,0],\"y\":[-3,-12]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[26, -1], [-1, 2]], [2, 0], [-3, -12], -132)","explanation_zh":"Wave4第二層核對=([[26, -1], [-1, 2]], [2, 0], [-3, -12], -132)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-evaluate","seed":202610284025,"parameters":{"matrix":[[26,-1],[-1,2]],"x":[2,0],"y":[-3,-12],"domain":"Exact_R2_matrix_inner_product","operation":"inner-evaluate","semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[26,-1],[-1,2]],\"x\":[2,0],\"y\":[-3,-12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0026","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[2,-1],[0,2]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[2, -1], [0, 2]], [-1, 0], 否)","explanation_zh":"Wave4第二層核對=([[2, -1], [0, 2]], [-1, 0], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284026,"parameters":{"matrix":[[2,-1],[0,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[2,-1],[0,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0027","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[3,0],[0,1]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[3, 0], [0, 1]], [0, 0], 是)","explanation_zh":"Wave4第二層核對=([[3, 0], [0, 1]], [0, 0], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284027,"parameters":{"matrix":[[3,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[3,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0028","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[4,1],[1,2]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[4, 1], [1, 2]], [1, 1], 是)","explanation_zh":"Wave4第二層核對=([[4, 1], [1, 2]], [1, 1], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284028,"parameters":{"matrix":[[4,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[4,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0029","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[5,2],[2,5]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[5, 2], [2, 5]], [2, 2], 是)","explanation_zh":"Wave4第二層核對=([[5, 2], [2, 5]], [2, 2], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284029,"parameters":{"matrix":[[5,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[5,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0030","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[6,-1],[-1,2]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[6, -1], [-1, 2]], [-1, -1], 是)","explanation_zh":"Wave4第二層核對=([[6, -1], [-1, 2]], [-1, -1], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284030,"parameters":{"matrix":[[6,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[6,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0031","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[7,0],[1,1]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[7, 0], [1, 1]], [0, 1], 否)","explanation_zh":"Wave4第二層核對=([[7, 0], [1, 1]], [0, 1], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284031,"parameters":{"matrix":[[7,0],[1,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[7,0],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0032","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[8,1],[1,2]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[8, 1], [1, 2]], [1, 1], 是)","explanation_zh":"Wave4第二層核對=([[8, 1], [1, 2]], [1, 1], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284032,"parameters":{"matrix":[[8,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[8,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0033","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[9,2],[2,5]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[9, 2], [2, 5]], [2, 2], 是)","explanation_zh":"Wave4第二層核對=([[9, 2], [2, 5]], [2, 2], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284033,"parameters":{"matrix":[[9,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[9,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0034","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[10,-1],[-1,2]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[10, -1], [-1, 2]], [-1, -1], 是)","explanation_zh":"Wave4第二層核對=([[10, -1], [-1, 2]], [-1, -1], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284034,"parameters":{"matrix":[[10,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[10,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0035","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[11,0],[0,1]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[11, 0], [0, 1]], [0, 0], 是)","explanation_zh":"Wave4第二層核對=([[11, 0], [0, 1]], [0, 0], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284035,"parameters":{"matrix":[[11,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[11,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0036","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[12,1],[2,2]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[12, 1], [2, 2]], [1, 2], 否)","explanation_zh":"Wave4第二層核對=([[12, 1], [2, 2]], [1, 2], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284036,"parameters":{"matrix":[[12,1],[2,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[12,1],[2,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0037","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[13,2],[2,5]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[13, 2], [2, 5]], [2, 2], 是)","explanation_zh":"Wave4第二層核對=([[13, 2], [2, 5]], [2, 2], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284037,"parameters":{"matrix":[[13,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[13,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0038","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[14,-1],[-1,2]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[14, -1], [-1, 2]], [-1, -1], 是)","explanation_zh":"Wave4第二層核對=([[14, -1], [-1, 2]], [-1, -1], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284038,"parameters":{"matrix":[[14,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[14,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0039","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[15,0],[0,1]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[15, 0], [0, 1]], [0, 0], 是)","explanation_zh":"Wave4第二層核對=([[15, 0], [0, 1]], [0, 0], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284039,"parameters":{"matrix":[[15,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[15,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0040","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[16,1],[1,2]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[16, 1], [1, 2]], [1, 1], 是)","explanation_zh":"Wave4第二層核對=([[16, 1], [1, 2]], [1, 1], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284040,"parameters":{"matrix":[[16,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[16,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0041","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[17,2],[3,5]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[17, 2], [3, 5]], [2, 3], 否)","explanation_zh":"Wave4第二層核對=([[17, 2], [3, 5]], [2, 3], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284041,"parameters":{"matrix":[[17,2],[3,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[17,2],[3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0042","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[18,-1],[-1,2]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[18, -1], [-1, 2]], [-1, -1], 是)","explanation_zh":"Wave4第二層核對=([[18, -1], [-1, 2]], [-1, -1], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284042,"parameters":{"matrix":[[18,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[18,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0043","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[19,0],[0,1]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[19, 0], [0, 1]], [0, 0], 是)","explanation_zh":"Wave4第二層核對=([[19, 0], [0, 1]], [0, 0], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284043,"parameters":{"matrix":[[19,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[19,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0044","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[20,1],[1,2]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[20, 1], [1, 2]], [1, 1], 是)","explanation_zh":"Wave4第二層核對=([[20, 1], [1, 2]], [1, 1], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284044,"parameters":{"matrix":[[20,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[20,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0045","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[21,2],[2,5]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[21, 2], [2, 5]], [2, 2], 是)","explanation_zh":"Wave4第二層核對=([[21, 2], [2, 5]], [2, 2], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284045,"parameters":{"matrix":[[21,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[21,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0046","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[22,-1],[0,2]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[22, -1], [0, 2]], [-1, 0], 否)","explanation_zh":"Wave4第二層核對=([[22, -1], [0, 2]], [-1, 0], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284046,"parameters":{"matrix":[[22,-1],[0,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[22,-1],[0,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0047","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[23,0],[0,1]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[23, 0], [0, 1]], [0, 0], 是)","explanation_zh":"Wave4第二層核對=([[23, 0], [0, 1]], [0, 0], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284047,"parameters":{"matrix":[[23,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[23,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0048","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[24,1],[1,2]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[24, 1], [1, 2]], [1, 1], 是)","explanation_zh":"Wave4第二層核對=([[24, 1], [1, 2]], [1, 1], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284048,"parameters":{"matrix":[[24,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[24,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0049","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[25,2],[2,5]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[25, 2], [2, 5]], [2, 2], 是)","explanation_zh":"Wave4第二層核對=([[25, 2], [2, 5]], [2, 2], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284049,"parameters":{"matrix":[[25,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[25,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0050","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-symmetry；資料={\"matrix\":[[26,-1],[-1,2]]}。請依 [matrix,off_diagonal_pair,symmetric] 欄序回答。","answer":{"kind":"expression","value":"([[26, -1], [-1, 2]], [-1, -1], 是)","explanation_zh":"Wave4第二層核對=([[26, -1], [-1, 2]], [-1, -1], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-symmetry","seed":202610284050,"parameters":{"matrix":[[26,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-symmetry","semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[26,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0051","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[1,0],[0,0]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 0]], 是, 1, 0, 否, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[1, 0], [0, 0]], 是, 1, 0, 否, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284051,"parameters":{"matrix":[[1,0],[0,0]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[1,0],[0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0052","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[3,0],[0,1]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[3, 0], [0, 1]], 是, 3, 3, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[3, 0], [0, 1]], 是, 3, 3, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284052,"parameters":{"matrix":[[3,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[3,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0053","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[4,1],[1,2]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[4, 1], [1, 2]], 是, 4, 7, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[4, 1], [1, 2]], 是, 4, 7, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284053,"parameters":{"matrix":[[4,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[4,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0054","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[5,2],[2,5]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[5, 2], [2, 5]], 是, 5, 21, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[5, 2], [2, 5]], 是, 5, 21, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284054,"parameters":{"matrix":[[5,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[5,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0055","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[6,-1],[-1,2]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[6, -1], [-1, 2]], 是, 6, 11, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[6, -1], [-1, 2]], 是, 6, 11, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284055,"parameters":{"matrix":[[6,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[6,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0056","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[7,0],[0,1]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[7, 0], [0, 1]], 是, 7, 7, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[7, 0], [0, 1]], 是, 7, 7, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284056,"parameters":{"matrix":[[7,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[7,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0057","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[8,1],[1,2]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[8, 1], [1, 2]], 是, 8, 15, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[8, 1], [1, 2]], 是, 8, 15, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284057,"parameters":{"matrix":[[8,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[8,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0058","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[8,0],[0,0]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[8, 0], [0, 0]], 是, 8, 0, 否, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[8, 0], [0, 0]], 是, 8, 0, 否, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284058,"parameters":{"matrix":[[8,0],[0,0]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[8,0],[0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0059","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[10,-1],[-1,2]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[10, -1], [-1, 2]], 是, 10, 19, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[10, -1], [-1, 2]], 是, 10, 19, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284059,"parameters":{"matrix":[[10,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[10,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0060","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[11,0],[0,1]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[11, 0], [0, 1]], 是, 11, 11, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[11, 0], [0, 1]], 是, 11, 11, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284060,"parameters":{"matrix":[[11,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[11,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0061","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[12,1],[1,2]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[12, 1], [1, 2]], 是, 12, 23, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[12, 1], [1, 2]], 是, 12, 23, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284061,"parameters":{"matrix":[[12,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[12,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0062","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[13,2],[2,5]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[13, 2], [2, 5]], 是, 13, 61, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[13, 2], [2, 5]], 是, 13, 61, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284062,"parameters":{"matrix":[[13,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[13,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0063","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[14,-1],[-1,2]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[14, -1], [-1, 2]], 是, 14, 27, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[14, -1], [-1, 2]], 是, 14, 27, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284063,"parameters":{"matrix":[[14,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[14,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0064","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[15,0],[0,1]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[15, 0], [0, 1]], 是, 15, 15, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[15, 0], [0, 1]], 是, 15, 15, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284064,"parameters":{"matrix":[[15,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[15,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0065","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[15,0],[0,0]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[15, 0], [0, 0]], 是, 15, 0, 否, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[15, 0], [0, 0]], 是, 15, 0, 否, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284065,"parameters":{"matrix":[[15,0],[0,0]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[15,0],[0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0066","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[17,2],[2,5]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[17, 2], [2, 5]], 是, 17, 81, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[17, 2], [2, 5]], 是, 17, 81, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284066,"parameters":{"matrix":[[17,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[17,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0067","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[18,-1],[-1,2]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[18, -1], [-1, 2]], 是, 18, 35, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[18, -1], [-1, 2]], 是, 18, 35, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284067,"parameters":{"matrix":[[18,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[18,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0068","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[19,0],[0,1]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[19, 0], [0, 1]], 是, 19, 19, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[19, 0], [0, 1]], 是, 19, 19, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284068,"parameters":{"matrix":[[19,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[19,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0069","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[20,1],[1,2]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[20, 1], [1, 2]], 是, 20, 39, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[20, 1], [1, 2]], 是, 20, 39, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284069,"parameters":{"matrix":[[20,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[20,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0070","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[21,2],[2,5]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[21, 2], [2, 5]], 是, 21, 101, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[21, 2], [2, 5]], 是, 21, 101, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284070,"parameters":{"matrix":[[21,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[21,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0071","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[22,-1],[-1,2]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[22, -1], [-1, 2]], 是, 22, 43, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[22, -1], [-1, 2]], 是, 22, 43, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284071,"parameters":{"matrix":[[22,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[22,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0072","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[22,0],[0,0]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[22, 0], [0, 0]], 是, 22, 0, 否, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[22, 0], [0, 0]], 是, 22, 0, 否, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284072,"parameters":{"matrix":[[22,0],[0,0]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[22,0],[0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0073","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[24,1],[1,2]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[24, 1], [1, 2]], 是, 24, 47, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[24, 1], [1, 2]], 是, 24, 47, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284073,"parameters":{"matrix":[[24,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[24,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0074","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[25,2],[2,5]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[25, 2], [2, 5]], 是, 25, 121, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[25, 2], [2, 5]], 是, 25, 121, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284074,"parameters":{"matrix":[[25,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[25,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0075","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-positive-definite；資料={\"matrix\":[[26,-1],[-1,2]]}。請依 [matrix,symmetric,leading_minor,determinant,positive_definite,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[26, -1], [-1, 2]], 是, 26, 51, 是, Sylvester_R2_universal_certificate)","explanation_zh":"Wave4第二層核對=([[26, -1], [-1, 2]], 是, 26, 51, 是, Sylvester_R2_universal_certificate)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-positive-definite","seed":202610284075,"parameters":{"matrix":[[26,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-positive-definite","semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[26,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0076","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[1,0],[0,0]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 0]], 是, 是, 否, 否, not_inner_product)","explanation_zh":"Wave4第二層核對=([[1, 0], [0, 0]], 是, 是, 否, 否, not_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284076,"parameters":{"matrix":[[1,0],[0,0]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[1,0],[0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0077","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[3,0],[0,1]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[3, 0], [0, 1]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[3, 0], [0, 1]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284077,"parameters":{"matrix":[[3,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[3,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0078","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[4,1],[1,2]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[4, 1], [1, 2]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[4, 1], [1, 2]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284078,"parameters":{"matrix":[[4,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[4,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0079","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[5,2],[2,5]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[5, 2], [2, 5]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[5, 2], [2, 5]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284079,"parameters":{"matrix":[[5,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[5,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0080","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[6,-1],[-1,2]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[6, -1], [-1, 2]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[6, -1], [-1, 2]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284080,"parameters":{"matrix":[[6,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[6,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0081","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[7,0],[1,1]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[7, 0], [1, 1]], 是, 否, 否, 否, not_inner_product)","explanation_zh":"Wave4第二層核對=([[7, 0], [1, 1]], 是, 否, 否, 否, not_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284081,"parameters":{"matrix":[[7,0],[1,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[7,0],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0082","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[8,1],[1,2]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[8, 1], [1, 2]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[8, 1], [1, 2]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284082,"parameters":{"matrix":[[8,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[8,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0083","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[8,0],[0,0]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[8, 0], [0, 0]], 是, 是, 否, 否, not_inner_product)","explanation_zh":"Wave4第二層核對=([[8, 0], [0, 0]], 是, 是, 否, 否, not_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284083,"parameters":{"matrix":[[8,0],[0,0]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[8,0],[0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0084","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[10,-1],[-1,2]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[10, -1], [-1, 2]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[10, -1], [-1, 2]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284084,"parameters":{"matrix":[[10,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[10,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0085","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[11,0],[0,1]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[11, 0], [0, 1]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[11, 0], [0, 1]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284085,"parameters":{"matrix":[[11,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[11,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0086","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[12,1],[2,2]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[12, 1], [2, 2]], 是, 否, 否, 否, not_inner_product)","explanation_zh":"Wave4第二層核對=([[12, 1], [2, 2]], 是, 否, 否, 否, not_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284086,"parameters":{"matrix":[[12,1],[2,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[12,1],[2,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0087","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[13,2],[2,5]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[13, 2], [2, 5]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[13, 2], [2, 5]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284087,"parameters":{"matrix":[[13,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[13,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0088","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[14,-1],[-1,2]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[14, -1], [-1, 2]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[14, -1], [-1, 2]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284088,"parameters":{"matrix":[[14,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[14,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0089","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[15,0],[0,1]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[15, 0], [0, 1]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[15, 0], [0, 1]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284089,"parameters":{"matrix":[[15,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[15,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0090","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[15,0],[0,0]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[15, 0], [0, 0]], 是, 是, 否, 否, not_inner_product)","explanation_zh":"Wave4第二層核對=([[15, 0], [0, 0]], 是, 是, 否, 否, not_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284090,"parameters":{"matrix":[[15,0],[0,0]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[15,0],[0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0091","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[17,2],[3,5]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[17, 2], [3, 5]], 是, 否, 否, 否, not_inner_product)","explanation_zh":"Wave4第二層核對=([[17, 2], [3, 5]], 是, 否, 否, 否, not_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284091,"parameters":{"matrix":[[17,2],[3,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[17,2],[3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0092","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[18,-1],[-1,2]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[18, -1], [-1, 2]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[18, -1], [-1, 2]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284092,"parameters":{"matrix":[[18,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[18,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0093","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[19,0],[0,1]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[19, 0], [0, 1]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[19, 0], [0, 1]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284093,"parameters":{"matrix":[[19,0],[0,1]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[19,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0094","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[20,1],[1,2]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[20, 1], [1, 2]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[20, 1], [1, 2]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284094,"parameters":{"matrix":[[20,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[20,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0095","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[21,2],[2,5]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[21, 2], [2, 5]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[21, 2], [2, 5]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284095,"parameters":{"matrix":[[21,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[21,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0096","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[22,-1],[0,2]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[22, -1], [0, 2]], 是, 否, 否, 否, not_inner_product)","explanation_zh":"Wave4第二層核對=([[22, -1], [0, 2]], 是, 否, 否, 否, not_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284096,"parameters":{"matrix":[[22,-1],[0,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[22,-1],[0,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0097","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[22,0],[0,0]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[22, 0], [0, 0]], 是, 是, 否, 否, not_inner_product)","explanation_zh":"Wave4第二層核對=([[22, 0], [0, 0]], 是, 是, 否, 否, not_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284097,"parameters":{"matrix":[[22,0],[0,0]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[22,0],[0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0098","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[24,1],[1,2]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[24, 1], [1, 2]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[24, 1], [1, 2]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284098,"parameters":{"matrix":[[24,1],[1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[24,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0099","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[25,2],[2,5]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[25, 2], [2, 5]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[25, 2], [2, 5]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284099,"parameters":{"matrix":[[25,2],[2,5]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[25,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r04-real-inner-product-space-0100","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-inner-product-space"],"title_zh":"實內積空間","stem_zh":"內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[26,-1],[-1,2]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。","answer":{"kind":"expression","value":"([[26, -1], [-1, 2]], 是, 是, 是, 是, real_inner_product)","explanation_zh":"Wave4第二層核對=([[26, -1], [-1, 2]], 是, 是, 是, 是, real_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實內積空間"]},"variant":{"family_id":"qf-mod28-inner-qualification","seed":202610284100,"parameters":{"matrix":[[26,-1],[-1,2]],"domain":"Exact_R2_matrix_inner_product","operation":"inner-qualification","semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[26,-1],[-1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}