{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0001","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[2,0],[0,1]],\"x\":[-3,-2]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[2, 0], [0, 1]], [-3, -2], 22, sqrt(22), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[2, 0], [0, 1]], [-3, -2], 22, sqrt(22), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293001,"parameters":{"matrix":[[2,0],[0,1]],"x":[-3,-2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[2,0],[0,1]],\"x\":[-3,-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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0002","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[3,1],[1,2]],\"x\":[-2,-1]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[3, 1], [1, 2]], [-2, -1], 18, sqrt(18), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[3, 1], [1, 2]], [-2, -1], 18, sqrt(18), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293002,"parameters":{"matrix":[[3,1],[1,2]],"x":[-2,-1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[3,1],[1,2]],\"x\":[-2,-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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0003","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[4,2],[2,5]],\"x\":[-1,0]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[4, 2], [2, 5]], [-1, 0], 4, sqrt(4), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[4, 2], [2, 5]], [-1, 0], 4, sqrt(4), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293003,"parameters":{"matrix":[[4,2],[2,5]],"x":[-1,0],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[4,2],[2,5]],\"x\":[-1,0]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0004","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[5,0],[0,1]],\"x\":[0,1]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[5, 0], [0, 1]], [0, 1], 1, sqrt(1), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[5, 0], [0, 1]], [0, 1], 1, sqrt(1), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293004,"parameters":{"matrix":[[5,0],[0,1]],"x":[0,1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[5,0],[0,1]],\"x\":[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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0005","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[6,1],[1,2]],\"x\":[1,2]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[6, 1], [1, 2]], [1, 2], 18, sqrt(18), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[6, 1], [1, 2]], [1, 2], 18, sqrt(18), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293005,"parameters":{"matrix":[[6,1],[1,2]],"x":[1,2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[6,1],[1,2]],\"x\":[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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0006","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[7,2],[2,5]],\"x\":[2,-2]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[7, 2], [2, 5]], [2, -2], 32, sqrt(32), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[7, 2], [2, 5]], [2, -2], 32, sqrt(32), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293006,"parameters":{"matrix":[[7,2],[2,5]],"x":[2,-2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[7,2],[2,5]],\"x\":[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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0007","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[8,0],[0,1]],\"x\":[3,-1]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[8, 0], [0, 1]], [3, -1], 73, sqrt(73), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[8, 0], [0, 1]], [3, -1], 73, sqrt(73), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293007,"parameters":{"matrix":[[8,0],[0,1]],"x":[3,-1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[8,0],[0,1]],\"x\":[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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0008","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[9,1],[1,2]],\"x\":[-3,0]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[9, 1], [1, 2]], [-3, 0], 81, sqrt(81), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[9, 1], [1, 2]], [-3, 0], 81, sqrt(81), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293008,"parameters":{"matrix":[[9,1],[1,2]],"x":[-3,0],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[9,1],[1,2]],\"x\":[-3,0]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0009","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[10,2],[2,5]],\"x\":[-2,1]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[10, 2], [2, 5]], [-2, 1], 37, sqrt(37), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[10, 2], [2, 5]], [-2, 1], 37, sqrt(37), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293009,"parameters":{"matrix":[[10,2],[2,5]],"x":[-2,1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[10,2],[2,5]],\"x\":[-2,1]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0010","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[11,0],[0,1]],\"x\":[-1,2]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[11, 0], [0, 1]], [-1, 2], 15, sqrt(15), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[11, 0], [0, 1]], [-1, 2], 15, sqrt(15), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293010,"parameters":{"matrix":[[11,0],[0,1]],"x":[-1,2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[11,0],[0,1]],\"x\":[-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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0011","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[12,1],[1,2]],\"x\":[0,-2]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[12, 1], [1, 2]], [0, -2], 8, sqrt(8), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[12, 1], [1, 2]], [0, -2], 8, sqrt(8), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293011,"parameters":{"matrix":[[12,1],[1,2]],"x":[0,-2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[12,1],[1,2]],\"x\":[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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0012","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[13,2],[2,5]],\"x\":[1,-1]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[13, 2], [2, 5]], [1, -1], 14, sqrt(14), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[13, 2], [2, 5]], [1, -1], 14, sqrt(14), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293012,"parameters":{"matrix":[[13,2],[2,5]],"x":[1,-1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[13,2],[2,5]],\"x\":[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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0013","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[14,0],[0,1]],\"x\":[2,0]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[14, 0], [0, 1]], [2, 0], 56, sqrt(56), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[14, 0], [0, 1]], [2, 0], 56, sqrt(56), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293013,"parameters":{"matrix":[[14,0],[0,1]],"x":[2,0],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[14,0],[0,1]],\"x\":[2,0]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0014","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[15,1],[1,2]],\"x\":[3,1]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[15, 1], [1, 2]], [3, 1], 143, sqrt(143), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[15, 1], [1, 2]], [3, 1], 143, sqrt(143), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293014,"parameters":{"matrix":[[15,1],[1,2]],"x":[3,1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[15,1],[1,2]],\"x\":[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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0015","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[16,2],[2,5]],\"x\":[-3,2]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[16, 2], [2, 5]], [-3, 2], 140, sqrt(140), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[16, 2], [2, 5]], [-3, 2], 140, sqrt(140), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293015,"parameters":{"matrix":[[16,2],[2,5]],"x":[-3,2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[16,2],[2,5]],\"x\":[-3,2]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0016","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[17,0],[0,1]],\"x\":[-2,-2]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[17, 0], [0, 1]], [-2, -2], 72, sqrt(72), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[17, 0], [0, 1]], [-2, -2], 72, sqrt(72), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293016,"parameters":{"matrix":[[17,0],[0,1]],"x":[-2,-2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[17,0],[0,1]],\"x\":[-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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0017","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[18,1],[1,2]],\"x\":[-1,-1]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[18, 1], [1, 2]], [-1, -1], 22, sqrt(22), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[18, 1], [1, 2]], [-1, -1], 22, sqrt(22), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293017,"parameters":{"matrix":[[18,1],[1,2]],"x":[-1,-1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[18,1],[1,2]],\"x\":[-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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0018","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[19,2],[2,5]],\"x\":[0,0]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[19, 2], [2, 5]], [0, 0], 0, sqrt(0), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[19, 2], [2, 5]], [0, 0], 0, sqrt(0), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293018,"parameters":{"matrix":[[19,2],[2,5]],"x":[0,0],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[19,2],[2,5]],\"x\":[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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0019","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[20,0],[0,1]],\"x\":[1,1]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[20, 0], [0, 1]], [1, 1], 21, sqrt(21), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[20, 0], [0, 1]], [1, 1], 21, sqrt(21), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293019,"parameters":{"matrix":[[20,0],[0,1]],"x":[1,1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[20,0],[0,1]],\"x\":[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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0020","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[21,1],[1,2]],\"x\":[2,2]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[21, 1], [1, 2]], [2, 2], 100, sqrt(100), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[21, 1], [1, 2]], [2, 2], 100, sqrt(100), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293020,"parameters":{"matrix":[[21,1],[1,2]],"x":[2,2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[21,1],[1,2]],\"x\":[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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0021","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[22,2],[2,5]],\"x\":[3,-2]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[22, 2], [2, 5]], [3, -2], 194, sqrt(194), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[22, 2], [2, 5]], [3, -2], 194, sqrt(194), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293021,"parameters":{"matrix":[[22,2],[2,5]],"x":[3,-2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[22,2],[2,5]],\"x\":[3,-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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0022","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[23,0],[0,1]],\"x\":[-3,-1]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[23, 0], [0, 1]], [-3, -1], 208, sqrt(208), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[23, 0], [0, 1]], [-3, -1], 208, sqrt(208), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293022,"parameters":{"matrix":[[23,0],[0,1]],"x":[-3,-1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[23,0],[0,1]],\"x\":[-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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0023","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[24,1],[1,2]],\"x\":[-2,0]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[24, 1], [1, 2]], [-2, 0], 96, sqrt(96), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[24, 1], [1, 2]], [-2, 0], 96, sqrt(96), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293023,"parameters":{"matrix":[[24,1],[1,2]],"x":[-2,0],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[24,1],[1,2]],\"x\":[-2,0]}"}},"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0024","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[25,2],[2,5]],\"x\":[-1,1]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[25, 2], [2, 5]], [-1, 1], 26, sqrt(26), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[25, 2], [2, 5]], [-1, 1], 26, sqrt(26), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293024,"parameters":{"matrix":[[25,2],[2,5]],"x":[-1,1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[25,2],[2,5]],\"x\":[-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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0025","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-square；資料={\"matrix\":[[26,0],[0,1]],\"x\":[0,2]}。請依 [matrix,x,norm_square,exact_norm,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[26, 0], [0, 1]], [0, 2], 4, sqrt(4), same_inner_product_universal_norm_certificate)","explanation_zh":"Wave4第三層核對=([[26, 0], [0, 1]], [0, 2], 4, sqrt(4), same_inner_product_universal_norm_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-square","seed":202610293025,"parameters":{"matrix":[[26,0],[0,1]],"x":[0,2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-square","semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[26,0],[0,1]],\"x\":[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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0026","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[2,0],[0,1]],\"x\":[-3,-2],\"y\":[-5,12]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([-3, -2], [-5, 12], [2, -14], 204, sqrt(204))","explanation_zh":"Wave4第三層核對=([-3, -2], [-5, 12], [2, -14], 204, sqrt(204))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293026,"parameters":{"matrix":[[2,0],[0,1]],"x":[-3,-2],"y":[-5,12],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[2,0],[0,1]],\"x\":[-3,-2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0027","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[3,1],[1,2]],\"x\":[-2,-1],\"y\":[-4,11]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([-2, -1], [-4, 11], [2, -12], 252, sqrt(252))","explanation_zh":"Wave4第三層核對=([-2, -1], [-4, 11], [2, -12], 252, sqrt(252))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293027,"parameters":{"matrix":[[3,1],[1,2]],"x":[-2,-1],"y":[-4,11],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[3,1],[1,2]],\"x\":[-2,-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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0028","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[4,2],[2,5]],\"x\":[-1,0],\"y\":[-3,10]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([-1, 0], [-3, 10], [2, -10], 436, sqrt(436))","explanation_zh":"Wave4第三層核對=([-1, 0], [-3, 10], [2, -10], 436, sqrt(436))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293028,"parameters":{"matrix":[[4,2],[2,5]],"x":[-1,0],"y":[-3,10],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[4,2],[2,5]],\"x\":[-1,0],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0029","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[5,0],[0,1]],\"x\":[0,1],\"y\":[-2,9]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [-2, 9], [2, -8], 84, sqrt(84))","explanation_zh":"Wave4第三層核對=([0, 1], [-2, 9], [2, -8], 84, sqrt(84))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293029,"parameters":{"matrix":[[5,0],[0,1]],"x":[0,1],"y":[-2,9],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[5,0],[0,1]],\"x\":[0,1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0030","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[6,1],[1,2]],\"x\":[1,2],\"y\":[-1,8]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [-1, 8], [2, -6], 72, sqrt(72))","explanation_zh":"Wave4第三層核對=([1, 2], [-1, 8], [2, -6], 72, sqrt(72))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293030,"parameters":{"matrix":[[6,1],[1,2]],"x":[1,2],"y":[-1,8],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[6,1],[1,2]],\"x\":[1,2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0031","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[7,2],[2,5]],\"x\":[2,-2],\"y\":[0,7]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([2, -2], [0, 7], [2, -9], 361, sqrt(361))","explanation_zh":"Wave4第三層核對=([2, -2], [0, 7], [2, -9], 361, sqrt(361))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293031,"parameters":{"matrix":[[7,2],[2,5]],"x":[2,-2],"y":[0,7],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[7,2],[2,5]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0032","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[8,0],[0,1]],\"x\":[3,-1],\"y\":[1,6]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([3, -1], [1, 6], [2, -7], 81, sqrt(81))","explanation_zh":"Wave4第三層核對=([3, -1], [1, 6], [2, -7], 81, sqrt(81))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293032,"parameters":{"matrix":[[8,0],[0,1]],"x":[3,-1],"y":[1,6],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[8,0],[0,1]],\"x\":[3,-1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0033","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[9,1],[1,2]],\"x\":[-3,0],\"y\":[2,5]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([-3, 0], [2, 5], [-5, -5], 325, sqrt(325))","explanation_zh":"Wave4第三層核對=([-3, 0], [2, 5], [-5, -5], 325, sqrt(325))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293033,"parameters":{"matrix":[[9,1],[1,2]],"x":[-3,0],"y":[2,5],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[9,1],[1,2]],\"x\":[-3,0],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0034","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[10,2],[2,5]],\"x\":[-2,1],\"y\":[3,4]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([-2, 1], [3, 4], [-5, -3], 355, sqrt(355))","explanation_zh":"Wave4第三層核對=([-2, 1], [3, 4], [-5, -3], 355, sqrt(355))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293034,"parameters":{"matrix":[[10,2],[2,5]],"x":[-2,1],"y":[3,4],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[10,2],[2,5]],\"x\":[-2,1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0035","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[11,0],[0,1]],\"x\":[-1,2],\"y\":[4,3]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([-1, 2], [4, 3], [-5, -1], 276, sqrt(276))","explanation_zh":"Wave4第三層核對=([-1, 2], [4, 3], [-5, -1], 276, sqrt(276))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293035,"parameters":{"matrix":[[11,0],[0,1]],"x":[-1,2],"y":[4,3],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[11,0],[0,1]],\"x\":[-1,2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0036","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[12,1],[1,2]],\"x\":[0,-2],\"y\":[5,2]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([0, -2], [5, 2], [-5, -4], 372, sqrt(372))","explanation_zh":"Wave4第三層核對=([0, -2], [5, 2], [-5, -4], 372, sqrt(372))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293036,"parameters":{"matrix":[[12,1],[1,2]],"x":[0,-2],"y":[5,2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[12,1],[1,2]],\"x\":[0,-2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0037","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[13,2],[2,5]],\"x\":[1,-1],\"y\":[-5,1]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([1, -1], [-5, 1], [6, -2], 440, sqrt(440))","explanation_zh":"Wave4第三層核對=([1, -1], [-5, 1], [6, -2], 440, sqrt(440))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293037,"parameters":{"matrix":[[13,2],[2,5]],"x":[1,-1],"y":[-5,1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0038","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[14,0],[0,1]],\"x\":[2,0],\"y\":[-4,0]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([2, 0], [-4, 0], [6, 0], 504, sqrt(504))","explanation_zh":"Wave4第三層核對=([2, 0], [-4, 0], [6, 0], 504, sqrt(504))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293038,"parameters":{"matrix":[[14,0],[0,1]],"x":[2,0],"y":[-4,0],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[14,0],[0,1]],\"x\":[2,0],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0039","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[15,1],[1,2]],\"x\":[3,1],\"y\":[-3,-1]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([3, 1], [-3, -1], [6, 2], 572, sqrt(572))","explanation_zh":"Wave4第三層核對=([3, 1], [-3, -1], [6, 2], 572, sqrt(572))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293039,"parameters":{"matrix":[[15,1],[1,2]],"x":[3,1],"y":[-3,-1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[15,1],[1,2]],\"x\":[3,1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0040","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[16,2],[2,5]],\"x\":[-3,2],\"y\":[-2,-2]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2], [-2, -2], [-1, 4], 80, sqrt(80))","explanation_zh":"Wave4第三層核對=([-3, 2], [-2, -2], [-1, 4], 80, sqrt(80))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293040,"parameters":{"matrix":[[16,2],[2,5]],"x":[-3,2],"y":[-2,-2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[16,2],[2,5]],\"x\":[-3,2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0041","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[17,0],[0,1]],\"x\":[-2,-2],\"y\":[-1,-3]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2], [-1, -3], [-1, 1], 18, sqrt(18))","explanation_zh":"Wave4第三層核對=([-2, -2], [-1, -3], [-1, 1], 18, sqrt(18))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293041,"parameters":{"matrix":[[17,0],[0,1]],"x":[-2,-2],"y":[-1,-3],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[17,0],[0,1]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0042","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[18,1],[1,2]],\"x\":[-1,-1],\"y\":[0,-4]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1], [0, -4], [-1, 3], 30, sqrt(30))","explanation_zh":"Wave4第三層核對=([-1, -1], [0, -4], [-1, 3], 30, sqrt(30))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293042,"parameters":{"matrix":[[18,1],[1,2]],"x":[-1,-1],"y":[0,-4],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0043","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[19,2],[2,5]],\"x\":[0,0],\"y\":[1,-5]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([0, 0], [1, -5], [-1, 5], 124, sqrt(124))","explanation_zh":"Wave4第三層核對=([0, 0], [1, -5], [-1, 5], 124, sqrt(124))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293043,"parameters":{"matrix":[[19,2],[2,5]],"x":[0,0],"y":[1,-5],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[19,2],[2,5]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0044","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[20,0],[0,1]],\"x\":[1,1],\"y\":[2,-6]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [2, -6], [-1, 7], 69, sqrt(69))","explanation_zh":"Wave4第三層核對=([1, 1], [2, -6], [-1, 7], 69, sqrt(69))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293044,"parameters":{"matrix":[[20,0],[0,1]],"x":[1,1],"y":[2,-6],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[20,0],[0,1]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0045","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[21,1],[1,2]],\"x\":[2,2],\"y\":[3,-7]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([2, 2], [3, -7], [-1, 9], 165, sqrt(165))","explanation_zh":"Wave4第三層核對=([2, 2], [3, -7], [-1, 9], 165, sqrt(165))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293045,"parameters":{"matrix":[[21,1],[1,2]],"x":[2,2],"y":[3,-7],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[21,1],[1,2]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0046","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[22,2],[2,5]],\"x\":[3,-2],\"y\":[4,-8]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([3, -2], [4, -8], [-1, 6], 178, sqrt(178))","explanation_zh":"Wave4第三層核對=([3, -2], [4, -8], [-1, 6], 178, sqrt(178))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293046,"parameters":{"matrix":[[22,2],[2,5]],"x":[3,-2],"y":[4,-8],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[22,2],[2,5]],\"x\":[3,-2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0047","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[23,0],[0,1]],\"x\":[-3,-1],\"y\":[5,-9]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([-3, -1], [5, -9], [-8, 8], 1536, sqrt(1536))","explanation_zh":"Wave4第三層核對=([-3, -1], [5, -9], [-8, 8], 1536, sqrt(1536))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293047,"parameters":{"matrix":[[23,0],[0,1]],"x":[-3,-1],"y":[5,-9],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[23,0],[0,1]],\"x\":[-3,-1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0048","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[24,1],[1,2]],\"x\":[-2,0],\"y\":[-5,-10]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([-2, 0], [-5, -10], [3, 10], 476, sqrt(476))","explanation_zh":"Wave4第三層核對=([-2, 0], [-5, -10], [3, 10], 476, sqrt(476))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293048,"parameters":{"matrix":[[24,1],[1,2]],"x":[-2,0],"y":[-5,-10],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[24,1],[1,2]],\"x\":[-2,0],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0049","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[25,2],[2,5]],\"x\":[-1,1],\"y\":[-4,-11]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([-1, 1], [-4, -11], [3, 12], 1089, sqrt(1089))","explanation_zh":"Wave4第三層核對=([-1, 1], [-4, -11], [3, 12], 1089, sqrt(1089))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293049,"parameters":{"matrix":[[25,2],[2,5]],"x":[-1,1],"y":[-4,-11],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0050","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={\"matrix\":[[26,0],[0,1]],\"x\":[0,2],\"y\":[-3,-12]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [-3, -12], [3, 14], 430, sqrt(430))","explanation_zh":"Wave4第三層核對=([0, 2], [-3, -12], [3, 14], 430, sqrt(430))；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-distance-square","seed":202610293050,"parameters":{"matrix":[[26,0],[0,1]],"x":[0,2],"y":[-3,-12],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-distance-square","semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[26,0],[0,1]],\"x\":[0,2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0051","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[2,0],[0,1]],\"x\":[-3,-2],\"y\":[-5,12]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(6, 22, 194, 36, 4268, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(6, 22, 194, 36, 4268, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293051,"parameters":{"matrix":[[2,0],[0,1]],"x":[-3,-2],"y":[-5,12],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[2,0],[0,1]],\"x\":[-3,-2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0052","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[3,1],[1,2]],\"x\":[-2,-1],\"y\":[-4,11]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-16, 18, 202, 256, 3636, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(-16, 18, 202, 256, 3636, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293052,"parameters":{"matrix":[[3,1],[1,2]],"x":[-2,-1],"y":[-4,11],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[3,1],[1,2]],\"x\":[-2,-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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0053","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[4,2],[2,5]],\"x\":[-1,0],\"y\":[-3,10]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-8, 4, 416, 64, 1664, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(-8, 4, 416, 64, 1664, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293053,"parameters":{"matrix":[[4,2],[2,5]],"x":[-1,0],"y":[-3,10],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[4,2],[2,5]],\"x\":[-1,0],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0054","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[5,0],[0,1]],\"x\":[0,1],\"y\":[-2,9]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(9, 1, 101, 81, 101, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(9, 1, 101, 81, 101, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293054,"parameters":{"matrix":[[5,0],[0,1]],"x":[0,1],"y":[-2,9],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[5,0],[0,1]],\"x\":[0,1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0055","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[6,1],[1,2]],\"x\":[1,2],\"y\":[-1,8]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(32, 18, 118, 1024, 2124, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(32, 18, 118, 1024, 2124, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293055,"parameters":{"matrix":[[6,1],[1,2]],"x":[1,2],"y":[-1,8],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[6,1],[1,2]],\"x\":[1,2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0056","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[7,2],[2,5]],\"x\":[2,-2],\"y\":[0,7]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-42, 32, 245, 1764, 7840, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(-42, 32, 245, 1764, 7840, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293056,"parameters":{"matrix":[[7,2],[2,5]],"x":[2,-2],"y":[0,7],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[7,2],[2,5]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0057","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[8,0],[0,1]],\"x\":[3,-1],\"y\":[1,6]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(18, 73, 44, 324, 3212, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(18, 73, 44, 324, 3212, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293057,"parameters":{"matrix":[[8,0],[0,1]],"x":[3,-1],"y":[1,6],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[8,0],[0,1]],\"x\":[3,-1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0058","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[9,1],[1,2]],\"x\":[-3,0],\"y\":[2,5]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-69, 81, 106, 4761, 8586, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(-69, 81, 106, 4761, 8586, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293058,"parameters":{"matrix":[[9,1],[1,2]],"x":[-3,0],"y":[2,5],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[9,1],[1,2]],\"x\":[-3,0],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0059","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[10,2],[2,5]],\"x\":[-2,1],\"y\":[3,4]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-50, 37, 218, 2500, 8066, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(-50, 37, 218, 2500, 8066, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293059,"parameters":{"matrix":[[10,2],[2,5]],"x":[-2,1],"y":[3,4],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[10,2],[2,5]],\"x\":[-2,1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0060","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[11,0],[0,1]],\"x\":[-1,2],\"y\":[4,3]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-38, 15, 185, 1444, 2775, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(-38, 15, 185, 1444, 2775, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293060,"parameters":{"matrix":[[11,0],[0,1]],"x":[-1,2],"y":[4,3],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[11,0],[0,1]],\"x\":[-1,2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0061","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[12,1],[1,2]],\"x\":[0,-2],\"y\":[5,2]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-18, 8, 328, 324, 2624, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(-18, 8, 328, 324, 2624, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293061,"parameters":{"matrix":[[12,1],[1,2]],"x":[0,-2],"y":[5,2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[12,1],[1,2]],\"x\":[0,-2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0062","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[13,2],[2,5]],\"x\":[1,-1],\"y\":[-5,1]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-58, 14, 310, 3364, 4340, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(-58, 14, 310, 3364, 4340, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293062,"parameters":{"matrix":[[13,2],[2,5]],"x":[1,-1],"y":[-5,1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0063","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[14,0],[0,1]],\"x\":[2,0],\"y\":[-4,0]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-112, 56, 224, 12544, 12544, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(-112, 56, 224, 12544, 12544, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293063,"parameters":{"matrix":[[14,0],[0,1]],"x":[2,0],"y":[-4,0],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[14,0],[0,1]],\"x\":[2,0],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0064","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[15,1],[1,2]],\"x\":[3,1],\"y\":[-3,-1]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-143, 143, 143, 20449, 20449, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(-143, 143, 143, 20449, 20449, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293064,"parameters":{"matrix":[[15,1],[1,2]],"x":[3,1],"y":[-3,-1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[15,1],[1,2]],\"x\":[3,1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0065","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[16,2],[2,5]],\"x\":[-3,2],\"y\":[-2,-2]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(80, 140, 100, 6400, 14000, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(80, 140, 100, 6400, 14000, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293065,"parameters":{"matrix":[[16,2],[2,5]],"x":[-3,2],"y":[-2,-2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[16,2],[2,5]],\"x\":[-3,2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0066","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[17,0],[0,1]],\"x\":[-2,-2],\"y\":[-1,-3]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(40, 72, 26, 1600, 1872, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(40, 72, 26, 1600, 1872, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293066,"parameters":{"matrix":[[17,0],[0,1]],"x":[-2,-2],"y":[-1,-3],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[17,0],[0,1]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0067","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[18,1],[1,2]],\"x\":[-1,-1],\"y\":[0,-4]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(12, 22, 32, 144, 704, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(12, 22, 32, 144, 704, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293067,"parameters":{"matrix":[[18,1],[1,2]],"x":[-1,-1],"y":[0,-4],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0068","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[19,2],[2,5]],\"x\":[0,0],\"y\":[1,-5]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 124, 0, 0, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(0, 0, 124, 0, 0, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293068,"parameters":{"matrix":[[19,2],[2,5]],"x":[0,0],"y":[1,-5],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[19,2],[2,5]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0069","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[20,0],[0,1]],\"x\":[1,1],\"y\":[2,-6]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(34, 21, 116, 1156, 2436, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(34, 21, 116, 1156, 2436, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293069,"parameters":{"matrix":[[20,0],[0,1]],"x":[1,1],"y":[2,-6],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[20,0],[0,1]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0070","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[21,1],[1,2]],\"x\":[2,2],\"y\":[3,-7]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(90, 100, 245, 8100, 24500, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(90, 100, 245, 8100, 24500, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293070,"parameters":{"matrix":[[21,1],[1,2]],"x":[2,2],"y":[3,-7],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[21,1],[1,2]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0071","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[22,2],[2,5]],\"x\":[3,-2],\"y\":[4,-8]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(280, 194, 544, 78400, 105536, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(280, 194, 544, 78400, 105536, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293071,"parameters":{"matrix":[[22,2],[2,5]],"x":[3,-2],"y":[4,-8],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[22,2],[2,5]],\"x\":[3,-2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0072","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[23,0],[0,1]],\"x\":[-3,-1],\"y\":[5,-9]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-336, 208, 656, 112896, 136448, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(-336, 208, 656, 112896, 136448, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293072,"parameters":{"matrix":[[23,0],[0,1]],"x":[-3,-1],"y":[5,-9],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[23,0],[0,1]],\"x\":[-3,-1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0073","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[24,1],[1,2]],\"x\":[-2,0],\"y\":[-5,-10]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(260, 96, 900, 67600, 86400, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(260, 96, 900, 67600, 86400, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293073,"parameters":{"matrix":[[24,1],[1,2]],"x":[-2,0],"y":[-5,-10],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[24,1],[1,2]],\"x\":[-2,0],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0074","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[25,2],[2,5]],\"x\":[-1,1],\"y\":[-4,-11]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(59, 26, 1181, 3481, 30706, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(59, 26, 1181, 3481, 30706, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293074,"parameters":{"matrix":[[25,2],[2,5]],"x":[-1,1],"y":[-4,-11],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0075","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={\"matrix\":[[26,0],[0,1]],\"x\":[0,2],\"y\":[-3,-12]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-24, 4, 378, 576, 1512, 是, Cauchy_Schwarz_universal_certificate)","explanation_zh":"Wave4第三層核對=(-24, 4, 378, 576, 1512, 是, Cauchy_Schwarz_universal_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293075,"parameters":{"matrix":[[26,0],[0,1]],"x":[0,2],"y":[-3,-12],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-cauchy-schwarz","semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[26,0],[0,1]],\"x\":[0,2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0076","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[2,0],[0,1]],\"x\":[-3,-2],\"y\":[-5,12]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([-3, -2], [-5, 12], 432, 432, 是)","explanation_zh":"Wave4第三層核對=([-3, -2], [-5, 12], 432, 432, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293076,"parameters":{"matrix":[[2,0],[0,1]],"x":[-3,-2],"y":[-5,12],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[2,0],[0,1]],\"x\":[-3,-2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0077","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[3,1],[1,2]],\"x\":[-2,-1],\"y\":[-4,11]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([-2, -1], [-4, 11], 440, 440, 是)","explanation_zh":"Wave4第三層核對=([-2, -1], [-4, 11], 440, 440, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293077,"parameters":{"matrix":[[3,1],[1,2]],"x":[-2,-1],"y":[-4,11],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[3,1],[1,2]],\"x\":[-2,-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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0078","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[4,2],[2,5]],\"x\":[-1,0],\"y\":[-3,10]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([-1, 0], [-3, 10], 840, 840, 是)","explanation_zh":"Wave4第三層核對=([-1, 0], [-3, 10], 840, 840, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293078,"parameters":{"matrix":[[4,2],[2,5]],"x":[-1,0],"y":[-3,10],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[4,2],[2,5]],\"x\":[-1,0],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0079","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[5,0],[0,1]],\"x\":[0,1],\"y\":[-2,9]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [-2, 9], 204, 204, 是)","explanation_zh":"Wave4第三層核對=([0, 1], [-2, 9], 204, 204, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293079,"parameters":{"matrix":[[5,0],[0,1]],"x":[0,1],"y":[-2,9],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[5,0],[0,1]],\"x\":[0,1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0080","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[6,1],[1,2]],\"x\":[1,2],\"y\":[-1,8]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [-1, 8], 272, 272, 是)","explanation_zh":"Wave4第三層核對=([1, 2], [-1, 8], 272, 272, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293080,"parameters":{"matrix":[[6,1],[1,2]],"x":[1,2],"y":[-1,8],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[6,1],[1,2]],\"x\":[1,2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0081","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[7,2],[2,5]],\"x\":[2,-2],\"y\":[0,7]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([2, -2], [0, 7], 554, 554, 是)","explanation_zh":"Wave4第三層核對=([2, -2], [0, 7], 554, 554, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293081,"parameters":{"matrix":[[7,2],[2,5]],"x":[2,-2],"y":[0,7],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[7,2],[2,5]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0082","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[8,0],[0,1]],\"x\":[3,-1],\"y\":[1,6]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([3, -1], [1, 6], 234, 234, 是)","explanation_zh":"Wave4第三層核對=([3, -1], [1, 6], 234, 234, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293082,"parameters":{"matrix":[[8,0],[0,1]],"x":[3,-1],"y":[1,6],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[8,0],[0,1]],\"x\":[3,-1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0083","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[9,1],[1,2]],\"x\":[-3,0],\"y\":[2,5]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([-3, 0], [2, 5], 374, 374, 是)","explanation_zh":"Wave4第三層核對=([-3, 0], [2, 5], 374, 374, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293083,"parameters":{"matrix":[[9,1],[1,2]],"x":[-3,0],"y":[2,5],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[9,1],[1,2]],\"x\":[-3,0],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0084","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[10,2],[2,5]],\"x\":[-2,1],\"y\":[3,4]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([-2, 1], [3, 4], 510, 510, 是)","explanation_zh":"Wave4第三層核對=([-2, 1], [3, 4], 510, 510, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293084,"parameters":{"matrix":[[10,2],[2,5]],"x":[-2,1],"y":[3,4],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[10,2],[2,5]],\"x\":[-2,1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0085","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[11,0],[0,1]],\"x\":[-1,2],\"y\":[4,3]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([-1, 2], [4, 3], 400, 400, 是)","explanation_zh":"Wave4第三層核對=([-1, 2], [4, 3], 400, 400, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293085,"parameters":{"matrix":[[11,0],[0,1]],"x":[-1,2],"y":[4,3],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[11,0],[0,1]],\"x\":[-1,2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0086","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[12,1],[1,2]],\"x\":[0,-2],\"y\":[5,2]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([0, -2], [5, 2], 672, 672, 是)","explanation_zh":"Wave4第三層核對=([0, -2], [5, 2], 672, 672, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293086,"parameters":{"matrix":[[12,1],[1,2]],"x":[0,-2],"y":[5,2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[12,1],[1,2]],\"x\":[0,-2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0087","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[13,2],[2,5]],\"x\":[1,-1],\"y\":[-5,1]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([1, -1], [-5, 1], 648, 648, 是)","explanation_zh":"Wave4第三層核對=([1, -1], [-5, 1], 648, 648, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293087,"parameters":{"matrix":[[13,2],[2,5]],"x":[1,-1],"y":[-5,1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0088","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[14,0],[0,1]],\"x\":[2,0],\"y\":[-4,0]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([2, 0], [-4, 0], 560, 560, 是)","explanation_zh":"Wave4第三層核對=([2, 0], [-4, 0], 560, 560, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293088,"parameters":{"matrix":[[14,0],[0,1]],"x":[2,0],"y":[-4,0],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[14,0],[0,1]],\"x\":[2,0],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0089","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[15,1],[1,2]],\"x\":[3,1],\"y\":[-3,-1]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([3, 1], [-3, -1], 572, 572, 是)","explanation_zh":"Wave4第三層核對=([3, 1], [-3, -1], 572, 572, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293089,"parameters":{"matrix":[[15,1],[1,2]],"x":[3,1],"y":[-3,-1],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[15,1],[1,2]],\"x\":[3,1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0090","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[16,2],[2,5]],\"x\":[-3,2],\"y\":[-2,-2]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2], [-2, -2], 480, 480, 是)","explanation_zh":"Wave4第三層核對=([-3, 2], [-2, -2], 480, 480, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293090,"parameters":{"matrix":[[16,2],[2,5]],"x":[-3,2],"y":[-2,-2],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[16,2],[2,5]],\"x\":[-3,2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0091","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[17,0],[0,1]],\"x\":[-2,-2],\"y\":[-1,-3]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2], [-1, -3], 196, 196, 是)","explanation_zh":"Wave4第三層核對=([-2, -2], [-1, -3], 196, 196, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293091,"parameters":{"matrix":[[17,0],[0,1]],"x":[-2,-2],"y":[-1,-3],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[17,0],[0,1]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0092","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[18,1],[1,2]],\"x\":[-1,-1],\"y\":[0,-4]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1], [0, -4], 108, 108, 是)","explanation_zh":"Wave4第三層核對=([-1, -1], [0, -4], 108, 108, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293092,"parameters":{"matrix":[[18,1],[1,2]],"x":[-1,-1],"y":[0,-4],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0093","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[19,2],[2,5]],\"x\":[0,0],\"y\":[1,-5]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([0, 0], [1, -5], 248, 248, 是)","explanation_zh":"Wave4第三層核對=([0, 0], [1, -5], 248, 248, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293093,"parameters":{"matrix":[[19,2],[2,5]],"x":[0,0],"y":[1,-5],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[19,2],[2,5]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0094","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[20,0],[0,1]],\"x\":[1,1],\"y\":[2,-6]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [2, -6], 274, 274, 是)","explanation_zh":"Wave4第三層核對=([1, 1], [2, -6], 274, 274, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293094,"parameters":{"matrix":[[20,0],[0,1]],"x":[1,1],"y":[2,-6],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[20,0],[0,1]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0095","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[21,1],[1,2]],\"x\":[2,2],\"y\":[3,-7]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([2, 2], [3, -7], 690, 690, 是)","explanation_zh":"Wave4第三層核對=([2, 2], [3, -7], 690, 690, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293095,"parameters":{"matrix":[[21,1],[1,2]],"x":[2,2],"y":[3,-7],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[21,1],[1,2]],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0096","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[22,2],[2,5]],\"x\":[3,-2],\"y\":[4,-8]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([3, -2], [4, -8], 1476, 1476, 是)","explanation_zh":"Wave4第三層核對=([3, -2], [4, -8], 1476, 1476, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293096,"parameters":{"matrix":[[22,2],[2,5]],"x":[3,-2],"y":[4,-8],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[22,2],[2,5]],\"x\":[3,-2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0097","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[23,0],[0,1]],\"x\":[-3,-1],\"y\":[5,-9]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([-3, -1], [5, -9], 1728, 1728, 是)","explanation_zh":"Wave4第三層核對=([-3, -1], [5, -9], 1728, 1728, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293097,"parameters":{"matrix":[[23,0],[0,1]],"x":[-3,-1],"y":[5,-9],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[23,0],[0,1]],\"x\":[-3,-1],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0098","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[24,1],[1,2]],\"x\":[-2,0],\"y\":[-5,-10]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([-2, 0], [-5, -10], 1992, 1992, 是)","explanation_zh":"Wave4第三層核對=([-2, 0], [-5, -10], 1992, 1992, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293098,"parameters":{"matrix":[[24,1],[1,2]],"x":[-2,0],"y":[-5,-10],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[24,1],[1,2]],\"x\":[-2,0],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0099","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[25,2],[2,5]],\"x\":[-1,1],\"y\":[-4,-11]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([-1, 1], [-4, -11], 2414, 2414, 是)","explanation_zh":"Wave4第三層核對=([-1, 1], [-4, -11], 2414, 2414, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293099,"parameters":{"matrix":[[25,2],[2,5]],"x":[-1,1],"y":[-4,-11],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0100","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","language":"zh-Hant","source_mko_ids":["mko-inner-product-induced-norm"],"title_zh":"內積誘導範數","stem_zh":"向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[26,0],[0,1]],\"x\":[0,2],\"y\":[-3,-12]}。請依 [x,y,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [-3, -12], 764, 764, 是)","explanation_zh":"Wave4第三層核對=([0, 2], [-3, -12], 764, 764, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["內積誘導範數"]},"variant":{"family_id":"qf-mod29-innernorm-parallelogram","seed":202610293100,"parameters":{"matrix":[[26,0],[0,1]],"x":[0,2],"y":[-3,-12],"domain":"Exact_R2_inner_product_induced_norm","operation":"innernorm-parallelogram","semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[26,0],[0,1]],\"x\":[0,2],\"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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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"}}