{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r04-real-bilinear-form-0001","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[-12,-3],[-5,-4]],\"x\":[-2,-3],\"y\":[-5,12]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[-12, -3], [-5, -4]], [-2, -3], [-5, 12], 21)","explanation_zh":"Wave4根節點核對=([[-12, -3], [-5, -4]], [-2, -3], [-5, 12], 21)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274001,"parameters":{"matrix":[[-12,-3],[-5,-4]],"x":[-2,-3],"y":[-5,12],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-12,-3],[-5,-4]],\"x\":[-2,-3],\"y\":[-5,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0002","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[-11,-2],[0,-3]],\"x\":[-1,-2],\"y\":[-4,11]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[-11, -2], [0, -3]], [-1, -2], [-4, 11], 44)","explanation_zh":"Wave4根節點核對=([[-11, -2], [0, -3]], [-1, -2], [-4, 11], 44)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274002,"parameters":{"matrix":[[-11,-2],[0,-3]],"x":[-1,-2],"y":[-4,11],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-11,-2],[0,-3]],\"x\":[-1,-2],\"y\":[-4,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0003","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[-10,-1],[5,-2]],\"x\":[0,-1],\"y\":[-3,10]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[-10, -1], [5, -2]], [0, -1], [-3, 10], 35)","explanation_zh":"Wave4根節點核對=([[-10, -1], [5, -2]], [0, -1], [-3, 10], 35)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274003,"parameters":{"matrix":[[-10,-1],[5,-2]],"x":[0,-1],"y":[-3,10],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-10,-1],[5,-2]],\"x\":[0,-1],\"y\":[-3,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0004","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[-9,0],[-1,-1]],\"x\":[1,0],\"y\":[-2,9]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[-9, 0], [-1, -1]], [1, 0], [-2, 9], 18)","explanation_zh":"Wave4根節點核對=([[-9, 0], [-1, -1]], [1, 0], [-2, 9], 18)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274004,"parameters":{"matrix":[[-9,0],[-1,-1]],"x":[1,0],"y":[-2,9],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-9,0],[-1,-1]],\"x\":[1,0],\"y\":[-2,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0005","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[-8,1],[4,0]],\"x\":[2,1],\"y\":[-1,8]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[-8, 1], [4, 0]], [2, 1], [-1, 8], 28)","explanation_zh":"Wave4根節點核對=([[-8, 1], [4, 0]], [2, 1], [-1, 8], 28)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274005,"parameters":{"matrix":[[-8,1],[4,0]],"x":[2,1],"y":[-1,8],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-8,1],[4,0]],\"x\":[2,1],\"y\":[-1,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0006","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[-7,2],[-2,1]],\"x\":[-2,2],\"y\":[0,7]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[-7, 2], [-2, 1]], [-2, 2], [0, 7], -14)","explanation_zh":"Wave4根節點核對=([[-7, 2], [-2, 1]], [-2, 2], [0, 7], -14)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274006,"parameters":{"matrix":[[-7,2],[-2,1]],"x":[-2,2],"y":[0,7],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-7,2],[-2,1]],\"x\":[-2,2],\"y\":[0,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0007","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[-6,3],[3,2]],\"x\":[-1,3],\"y\":[1,6]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[-6, 3], [3, 2]], [-1, 3], [1, 6], 33)","explanation_zh":"Wave4根節點核對=([[-6, 3], [3, 2]], [-1, 3], [1, 6], 33)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274007,"parameters":{"matrix":[[-6,3],[3,2]],"x":[-1,3],"y":[1,6],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-6,3],[3,2]],\"x\":[-1,3],\"y\":[1,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0008","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[-5,-3],[-3,3]],\"x\":[0,-3],\"y\":[2,5]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[-5, -3], [-3, 3]], [0, -3], [2, 5], -27)","explanation_zh":"Wave4根節點核對=([[-5, -3], [-3, 3]], [0, -3], [2, 5], -27)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274008,"parameters":{"matrix":[[-5,-3],[-3,3]],"x":[0,-3],"y":[2,5],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-5,-3],[-3,3]],\"x\":[0,-3],\"y\":[2,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0009","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[-4,-2],[2,4]],\"x\":[1,-2],\"y\":[3,4]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[-4, -2], [2, 4]], [1, -2], [3, 4], -64)","explanation_zh":"Wave4根節點核對=([[-4, -2], [2, 4]], [1, -2], [3, 4], -64)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274009,"parameters":{"matrix":[[-4,-2],[2,4]],"x":[1,-2],"y":[3,4],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-4,-2],[2,4]],\"x\":[1,-2],\"y\":[3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0010","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[-3,-1],[-4,-4]],\"x\":[2,-1],\"y\":[4,3]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[-3, -1], [-4, -4]], [2, -1], [4, 3], -2)","explanation_zh":"Wave4根節點核對=([[-3, -1], [-4, -4]], [2, -1], [4, 3], -2)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274010,"parameters":{"matrix":[[-3,-1],[-4,-4]],"x":[2,-1],"y":[4,3],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-3,-1],[-4,-4]],\"x\":[2,-1],\"y\":[4,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0011","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[-2,0],[1,-3]],\"x\":[-2,0],\"y\":[5,2]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[-2, 0], [1, -3]], [-2, 0], [5, 2], 20)","explanation_zh":"Wave4根節點核對=([[-2, 0], [1, -3]], [-2, 0], [5, 2], 20)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274011,"parameters":{"matrix":[[-2,0],[1,-3]],"x":[-2,0],"y":[5,2],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-2,0],[1,-3]],\"x\":[-2,0],\"y\":[5,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0012","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[-1,1],[-5,-2]],\"x\":[-1,1],\"y\":[-5,1]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[-1, 1], [-5, -2]], [-1, 1], [-5, 1], 17)","explanation_zh":"Wave4根節點核對=([[-1, 1], [-5, -2]], [-1, 1], [-5, 1], 17)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274012,"parameters":{"matrix":[[-1,1],[-5,-2]],"x":[-1,1],"y":[-5,1],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-1,1],[-5,-2]],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0013","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[0,2],[0,-1]],\"x\":[0,2],\"y\":[-4,0]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2], [0, -1]], [0, 2], [-4, 0], 0)","explanation_zh":"Wave4根節點核對=([[0, 2], [0, -1]], [0, 2], [-4, 0], 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274013,"parameters":{"matrix":[[0,2],[0,-1]],"x":[0,2],"y":[-4,0],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[0,2],[0,-1]],\"x\":[0,2],\"y\":[-4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0014","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[1,3],[5,0]],\"x\":[1,3],\"y\":[-3,-1]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [5, 0]], [1, 3], [-3, -1], -51)","explanation_zh":"Wave4根節點核對=([[1, 3], [5, 0]], [1, 3], [-3, -1], -51)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274014,"parameters":{"matrix":[[1,3],[5,0]],"x":[1,3],"y":[-3,-1],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[1,3],[5,0]],\"x\":[1,3],\"y\":[-3,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0015","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[2,-3],[-1,1]],\"x\":[2,-3],\"y\":[-2,-2]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[2, -3], [-1, 1]], [2, -3], [-2, -2], 4)","explanation_zh":"Wave4根節點核對=([[2, -3], [-1, 1]], [2, -3], [-2, -2], 4)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274015,"parameters":{"matrix":[[2,-3],[-1,1]],"x":[2,-3],"y":[-2,-2],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[2,-3],[-1,1]],\"x\":[2,-3],\"y\":[-2,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0016","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[3,-2],[4,2]],\"x\":[-2,-2],\"y\":[-1,-3]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[3, -2], [4, 2]], [-2, -2], [-1, -3], 14)","explanation_zh":"Wave4根節點核對=([[3, -2], [4, 2]], [-2, -2], [-1, -3], 14)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274016,"parameters":{"matrix":[[3,-2],[4,2]],"x":[-2,-2],"y":[-1,-3],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[3,-2],[4,2]],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0017","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[4,-1],[-2,3]],\"x\":[-1,-1],\"y\":[0,-4]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[4, -1], [-2, 3]], [-1, -1], [0, -4], 8)","explanation_zh":"Wave4根節點核對=([[4, -1], [-2, 3]], [-1, -1], [0, -4], 8)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274017,"parameters":{"matrix":[[4,-1],[-2,3]],"x":[-1,-1],"y":[0,-4],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[4,-1],[-2,3]],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0018","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[5,0],[3,4]],\"x\":[0,0],\"y\":[1,-5]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[5, 0], [3, 4]], [0, 0], [1, -5], 0)","explanation_zh":"Wave4根節點核對=([[5, 0], [3, 4]], [0, 0], [1, -5], 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274018,"parameters":{"matrix":[[5,0],[3,4]],"x":[0,0],"y":[1,-5],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[5,0],[3,4]],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0019","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[6,1],[-3,-4]],\"x\":[1,1],\"y\":[2,-6]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[6, 1], [-3, -4]], [1, 1], [2, -6], 24)","explanation_zh":"Wave4根節點核對=([[6, 1], [-3, -4]], [1, 1], [2, -6], 24)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274019,"parameters":{"matrix":[[6,1],[-3,-4]],"x":[1,1],"y":[2,-6],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[6,1],[-3,-4]],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0020","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[7,2],[2,-3]],\"x\":[2,2],\"y\":[3,-7]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[7, 2], [2, -3]], [2, 2], [3, -7], 68)","explanation_zh":"Wave4根節點核對=([[7, 2], [2, -3]], [2, 2], [3, -7], 68)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274020,"parameters":{"matrix":[[7,2],[2,-3]],"x":[2,2],"y":[3,-7],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[7,2],[2,-3]],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0021","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[8,3],[-4,-2]],\"x\":[-2,3],\"y\":[4,-8]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[8, 3], [-4, -2]], [-2, 3], [4, -8], -16)","explanation_zh":"Wave4根節點核對=([[8, 3], [-4, -2]], [-2, 3], [4, -8], -16)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274021,"parameters":{"matrix":[[8,3],[-4,-2]],"x":[-2,3],"y":[4,-8],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[8,3],[-4,-2]],\"x\":[-2,3],\"y\":[4,-8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0022","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[9,-3],[1,-1]],\"x\":[-1,-3],\"y\":[5,-9]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[9, -3], [1, -1]], [-1, -3], [5, -9], -114)","explanation_zh":"Wave4根節點核對=([[9, -3], [1, -1]], [-1, -3], [5, -9], -114)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274022,"parameters":{"matrix":[[9,-3],[1,-1]],"x":[-1,-3],"y":[5,-9],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[9,-3],[1,-1]],\"x\":[-1,-3],\"y\":[5,-9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0023","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[10,-2],[-5,0]],\"x\":[0,-2],\"y\":[-5,-10]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[10, -2], [-5, 0]], [0, -2], [-5, -10], -50)","explanation_zh":"Wave4根節點核對=([[10, -2], [-5, 0]], [0, -2], [-5, -10], -50)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274023,"parameters":{"matrix":[[10,-2],[-5,0]],"x":[0,-2],"y":[-5,-10],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[10,-2],[-5,0]],\"x\":[0,-2],\"y\":[-5,-10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0024","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[11,-1],[0,1]],\"x\":[1,-1],\"y\":[-4,-11]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[11, -1], [0, 1]], [1, -1], [-4, -11], -22)","explanation_zh":"Wave4根節點核對=([[11, -1], [0, 1]], [1, -1], [-4, -11], -22)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274024,"parameters":{"matrix":[[11,-1],[0,1]],"x":[1,-1],"y":[-4,-11],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[11,-1],[0,1]],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0025","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={\"matrix\":[[12,0],[5,2]],\"x\":[2,0],\"y\":[-3,-12]}。請依 [matrix,x,y,value] 欄序回答。","answer":{"kind":"expression","value":"([[12, 0], [5, 2]], [2, 0], [-3, -12], -72)","explanation_zh":"Wave4根節點核對=([[12, 0], [5, 2]], [2, 0], [-3, -12], -72)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274025,"parameters":{"matrix":[[12,0],[5,2]],"x":[2,0],"y":[-3,-12],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-matrix-evaluate","semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[12,0],[5,2]],\"x\":[2,0],\"y\":[-3,-12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0026","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[-12,-3],[-5,-4]],\"u\":[1,-2],\"v\":[-3,1],\"alpha\":-2,\"beta\":-3,\"y\":[-5,12]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, -3, [7, 1], 145, 145, 是)","explanation_zh":"Wave4根節點核對=(-2, -3, [7, 1], 145, 145, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274026,"parameters":{"matrix":[[-12,-3],[-5,-4]],"u":[1,-2],"v":[-3,1],"alpha":-2,"beta":-3,"y":[-5,12],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-2,\"beta\":-3,\"matrix\":[[-12,-3],[-5,-4]],\"u\":[1,-2],\"v\":[-3,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0027","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[-11,-2],[0,-3]],\"u\":[1,-1],\"v\":[-2,1],\"alpha\":-1,\"beta\":-2,\"y\":[-4,11]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, -2, [3, -1], 99, 99, 是)","explanation_zh":"Wave4根節點核對=(-1, -2, [3, -1], 99, 99, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274027,"parameters":{"matrix":[[-11,-2],[0,-3]],"u":[1,-1],"v":[-2,1],"alpha":-1,"beta":-2,"y":[-4,11],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-1,\"beta\":-2,\"matrix\":[[-11,-2],[0,-3]],\"u\":[1,-1],\"v\":[-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0028","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[-10,-1],[5,-2]],\"u\":[1,0],\"v\":[-1,1],\"alpha\":0,\"beta\":-1,\"y\":[-3,10]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, -1, [1, -1], 55, 55, 是)","explanation_zh":"Wave4根節點核對=(0, -1, [1, -1], 55, 55, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274028,"parameters":{"matrix":[[-10,-1],[5,-2]],"u":[1,0],"v":[-1,1],"alpha":0,"beta":-1,"y":[-3,10],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":0,\"beta\":-1,\"matrix\":[[-10,-1],[5,-2]],\"u\":[1,0],\"v\":[-1,1],\"y\":[-3,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0029","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[-9,0],[-1,-1]],\"u\":[1,1],\"v\":[0,1],\"alpha\":1,\"beta\":0,\"y\":[-2,9]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [1, 1], 11, 11, 是)","explanation_zh":"Wave4根節點核對=(1, 0, [1, 1], 11, 11, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274029,"parameters":{"matrix":[[-9,0],[-1,-1]],"u":[1,1],"v":[0,1],"alpha":1,"beta":0,"y":[-2,9],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":1,\"beta\":0,\"matrix\":[[-9,0],[-1,-1]],\"u\":[1,1],\"v\":[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0030","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[-8,1],[4,0]],\"u\":[1,2],\"v\":[1,1],\"alpha\":2,\"beta\":1,\"y\":[-1,8]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, [3, 5], 28, 28, 是)","explanation_zh":"Wave4根節點核對=(2, 1, [3, 5], 28, 28, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274030,"parameters":{"matrix":[[-8,1],[4,0]],"u":[1,2],"v":[1,1],"alpha":2,"beta":1,"y":[-1,8],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":2,\"beta\":1,\"matrix\":[[-8,1],[4,0]],\"u\":[1,2],\"v\":[1,1],\"y\":[-1,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0031","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[-7,2],[-2,1]],\"u\":[1,-2],\"v\":[2,1],\"alpha\":-2,\"beta\":2,\"y\":[0,7]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, 2, [2, 6], 70, 70, 是)","explanation_zh":"Wave4根節點核對=(-2, 2, [2, 6], 70, 70, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274031,"parameters":{"matrix":[[-7,2],[-2,1]],"u":[1,-2],"v":[2,1],"alpha":-2,"beta":2,"y":[0,7],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-2,\"beta\":2,\"matrix\":[[-7,2],[-2,1]],\"u\":[1,-2],\"v\":[2,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0032","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[-6,3],[3,2]],\"u\":[1,-1],\"v\":[3,1],\"alpha\":-1,\"beta\":3,\"y\":[1,6]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, 3, [8, 4], 156, 156, 是)","explanation_zh":"Wave4根節點核對=(-1, 3, [8, 4], 156, 156, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274032,"parameters":{"matrix":[[-6,3],[3,2]],"u":[1,-1],"v":[3,1],"alpha":-1,"beta":3,"y":[1,6],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-1,\"beta\":3,\"matrix\":[[-6,3],[3,2]],\"u\":[1,-1],\"v\":[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0033","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[-5,-3],[-3,3]],\"u\":[1,0],\"v\":[-3,1],\"alpha\":0,\"beta\":-3,\"y\":[2,5]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, -3, [9, -3], -252, -252, 是)","explanation_zh":"Wave4根節點核對=(0, -3, [9, -3], -252, -252, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274033,"parameters":{"matrix":[[-5,-3],[-3,3]],"u":[1,0],"v":[-3,1],"alpha":0,"beta":-3,"y":[2,5],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":0,\"beta\":-3,\"matrix\":[[-5,-3],[-3,3]],\"u\":[1,0],\"v\":[-3,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0034","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[-4,-2],[2,4]],\"u\":[1,1],\"v\":[-2,1],\"alpha\":1,\"beta\":-2,\"y\":[3,4]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, -2, [5, -1], -122, -122, 是)","explanation_zh":"Wave4根節點核對=(1, -2, [5, -1], -122, -122, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274034,"parameters":{"matrix":[[-4,-2],[2,4]],"u":[1,1],"v":[-2,1],"alpha":1,"beta":-2,"y":[3,4],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":1,\"beta\":-2,\"matrix\":[[-4,-2],[2,4]],\"u\":[1,1],\"v\":[-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0035","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[-3,-1],[-4,-4]],\"u\":[1,2],\"v\":[-1,1],\"alpha\":2,\"beta\":-1,\"y\":[4,3]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, -1, [3, 3], -129, -129, 是)","explanation_zh":"Wave4根節點核對=(2, -1, [3, 3], -129, -129, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274035,"parameters":{"matrix":[[-3,-1],[-4,-4]],"u":[1,2],"v":[-1,1],"alpha":2,"beta":-1,"y":[4,3],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":2,\"beta\":-1,\"matrix\":[[-3,-1],[-4,-4]],\"u\":[1,2],\"v\":[-1,1],\"y\":[4,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0036","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[-2,0],[1,-3]],\"u\":[1,-2],\"v\":[0,1],\"alpha\":-2,\"beta\":0,\"y\":[5,2]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, 0, [-2, 4], 16, 16, 是)","explanation_zh":"Wave4根節點核對=(-2, 0, [-2, 4], 16, 16, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274036,"parameters":{"matrix":[[-2,0],[1,-3]],"u":[1,-2],"v":[0,1],"alpha":-2,"beta":0,"y":[5,2],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-2,\"beta\":0,\"matrix\":[[-2,0],[1,-3]],\"u\":[1,-2],\"v\":[0,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0037","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[-1,1],[-5,-2]],\"u\":[1,-1],\"v\":[1,1],\"alpha\":-1,\"beta\":1,\"y\":[-5,1]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, 1, [0, 2], 46, 46, 是)","explanation_zh":"Wave4根節點核對=(-1, 1, [0, 2], 46, 46, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274037,"parameters":{"matrix":[[-1,1],[-5,-2]],"u":[1,-1],"v":[1,1],"alpha":-1,"beta":1,"y":[-5,1],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-1,\"beta\":1,\"matrix\":[[-1,1],[-5,-2]],\"u\":[1,-1],\"v\":[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0038","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[0,2],[0,-1]],\"u\":[1,0],\"v\":[2,1],\"alpha\":0,\"beta\":2,\"y\":[-4,0]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 2, [4, 2], 0, 0, 是)","explanation_zh":"Wave4根節點核對=(0, 2, [4, 2], 0, 0, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274038,"parameters":{"matrix":[[0,2],[0,-1]],"u":[1,0],"v":[2,1],"alpha":0,"beta":2,"y":[-4,0],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":0,\"beta\":2,\"matrix\":[[0,2],[0,-1]],\"u\":[1,0],\"v\":[2,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0039","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[1,3],[5,0]],\"u\":[1,1],\"v\":[3,1],\"alpha\":1,\"beta\":3,\"y\":[-3,-1]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, [10, 4], -120, -120, 是)","explanation_zh":"Wave4根節點核對=(1, 3, [10, 4], -120, -120, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274039,"parameters":{"matrix":[[1,3],[5,0]],"u":[1,1],"v":[3,1],"alpha":1,"beta":3,"y":[-3,-1],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":1,\"beta\":3,\"matrix\":[[1,3],[5,0]],\"u\":[1,1],\"v\":[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0040","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[2,-3],[-1,1]],\"u\":[1,2],\"v\":[-3,1],\"alpha\":2,\"beta\":-3,\"y\":[-2,-2]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, -3, [11, 1], 22, 22, 是)","explanation_zh":"Wave4根節點核對=(2, -3, [11, 1], 22, 22, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274040,"parameters":{"matrix":[[2,-3],[-1,1]],"u":[1,2],"v":[-3,1],"alpha":2,"beta":-3,"y":[-2,-2],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":2,\"beta\":-3,\"matrix\":[[2,-3],[-1,1]],\"u\":[1,2],\"v\":[-3,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0041","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[3,-2],[4,2]],\"u\":[1,-2],\"v\":[-2,1],\"alpha\":-2,\"beta\":-2,\"y\":[-1,-3]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, -2, [2, 2], -14, -14, 是)","explanation_zh":"Wave4根節點核對=(-2, -2, [2, 2], -14, -14, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274041,"parameters":{"matrix":[[3,-2],[4,2]],"u":[1,-2],"v":[-2,1],"alpha":-2,"beta":-2,"y":[-1,-3],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-2,\"beta\":-2,\"matrix\":[[3,-2],[4,2]],\"u\":[1,-2],\"v\":[-2,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0042","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[4,-1],[-2,3]],\"u\":[1,-1],\"v\":[-1,1],\"alpha\":-1,\"beta\":-1,\"y\":[0,-4]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, -1, [0, 0], 0, 0, 是)","explanation_zh":"Wave4根節點核對=(-1, -1, [0, 0], 0, 0, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274042,"parameters":{"matrix":[[4,-1],[-2,3]],"u":[1,-1],"v":[-1,1],"alpha":-1,"beta":-1,"y":[0,-4],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-1,\"beta\":-1,\"matrix\":[[4,-1],[-2,3]],\"u\":[1,-1],\"v\":[-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0043","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[5,0],[3,4]],\"u\":[1,0],\"v\":[0,1],\"alpha\":0,\"beta\":0,\"y\":[1,-5]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, [0, 0], 0, 0, 是)","explanation_zh":"Wave4根節點核對=(0, 0, [0, 0], 0, 0, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274043,"parameters":{"matrix":[[5,0],[3,4]],"u":[1,0],"v":[0,1],"alpha":0,"beta":0,"y":[1,-5],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":0,\"beta\":0,\"matrix\":[[5,0],[3,4]],\"u\":[1,0],\"v\":[0,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0044","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[6,1],[-3,-4]],\"u\":[1,1],\"v\":[1,1],\"alpha\":1,\"beta\":1,\"y\":[2,-6]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [2, 2], 48, 48, 是)","explanation_zh":"Wave4根節點核對=(1, 1, [2, 2], 48, 48, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274044,"parameters":{"matrix":[[6,1],[-3,-4]],"u":[1,1],"v":[1,1],"alpha":1,"beta":1,"y":[2,-6],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":1,\"beta\":1,\"matrix\":[[6,1],[-3,-4]],\"u\":[1,1],\"v\":[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0045","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[7,2],[2,-3]],\"u\":[1,2],\"v\":[2,1],\"alpha\":2,\"beta\":2,\"y\":[3,-7]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [6, 6], 204, 204, 是)","explanation_zh":"Wave4根節點核對=(2, 2, [6, 6], 204, 204, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274045,"parameters":{"matrix":[[7,2],[2,-3]],"u":[1,2],"v":[2,1],"alpha":2,"beta":2,"y":[3,-7],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":2,\"beta\":2,\"matrix\":[[7,2],[2,-3]],\"u\":[1,2],\"v\":[2,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0046","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[8,3],[-4,-2]],\"u\":[1,-2],\"v\":[3,1],\"alpha\":-2,\"beta\":3,\"y\":[4,-8]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, 3, [7, 7], 56, 56, 是)","explanation_zh":"Wave4根節點核對=(-2, 3, [7, 7], 56, 56, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274046,"parameters":{"matrix":[[8,3],[-4,-2]],"u":[1,-2],"v":[3,1],"alpha":-2,"beta":3,"y":[4,-8],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-2,\"beta\":3,\"matrix\":[[8,3],[-4,-2]],\"u\":[1,-2],\"v\":[3,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0047","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[9,-3],[1,-1]],\"u\":[1,-1],\"v\":[-3,1],\"alpha\":-1,\"beta\":-3,\"y\":[5,-9]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, -3, [8, -2], 548, 548, 是)","explanation_zh":"Wave4根節點核對=(-1, -3, [8, -2], 548, 548, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274047,"parameters":{"matrix":[[9,-3],[1,-1]],"u":[1,-1],"v":[-3,1],"alpha":-1,"beta":-3,"y":[5,-9],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-1,\"beta\":-3,\"matrix\":[[9,-3],[1,-1]],\"u\":[1,-1],\"v\":[-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0048","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[10,-2],[-5,0]],\"u\":[1,0],\"v\":[-2,1],\"alpha\":0,\"beta\":-2,\"y\":[-5,-10]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, -2, [4, -2], -170, -170, 是)","explanation_zh":"Wave4根節點核對=(0, -2, [4, -2], -170, -170, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274048,"parameters":{"matrix":[[10,-2],[-5,0]],"u":[1,0],"v":[-2,1],"alpha":0,"beta":-2,"y":[-5,-10],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":0,\"beta\":-2,\"matrix\":[[10,-2],[-5,0]],\"u\":[1,0],\"v\":[-2,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0049","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[11,-1],[0,1]],\"u\":[1,1],\"v\":[-1,1],\"alpha\":1,\"beta\":-1,\"y\":[-4,-11]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, -1, [2, 0], -66, -66, 是)","explanation_zh":"Wave4根節點核對=(1, -1, [2, 0], -66, -66, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274049,"parameters":{"matrix":[[11,-1],[0,1]],"u":[1,1],"v":[-1,1],"alpha":1,"beta":-1,"y":[-4,-11],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":1,\"beta\":-1,\"matrix\":[[11,-1],[0,1]],\"u\":[1,1],\"v\":[-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0050","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-first-slot-identity；資料={\"matrix\":[[12,0],[5,2]],\"u\":[1,2],\"v\":[0,1],\"alpha\":2,\"beta\":0,\"y\":[-3,-12]}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [2, 4], -228, -228, 是)","explanation_zh":"Wave4根節點核對=(2, 0, [2, 4], -228, -228, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274050,"parameters":{"matrix":[[12,0],[5,2]],"u":[1,2],"v":[0,1],"alpha":2,"beta":0,"y":[-3,-12],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-first-slot-identity","semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":2,\"beta\":0,\"matrix\":[[12,0],[5,2]],\"u\":[1,2],\"v\":[0,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0051","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-12,-3],[-5,-4]],\"x\":[-2,-3],\"u\":[1,-2],\"v\":[-3,1],\"alpha\":-2,\"beta\":-3}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, -3, [7, 1], 291, 291, 是)","explanation_zh":"Wave4根節點核對=(-2, -3, [7, 1], 291, 291, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274051,"parameters":{"matrix":[[-12,-3],[-5,-4]],"x":[-2,-3],"u":[1,-2],"v":[-3,1],"alpha":-2,"beta":-3,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-2,\"beta\":-3,\"matrix\":[[-12,-3],[-5,-4]],\"u\":[1,-2],\"v\":[-3,1],\"x\":[-2,-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0052","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-11,-2],[0,-3]],\"x\":[-1,-2],\"u\":[1,-1],\"v\":[-2,1],\"alpha\":-1,\"beta\":-2}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, -2, [3, -1], 25, 25, 是)","explanation_zh":"Wave4根節點核對=(-1, -2, [3, -1], 25, 25, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274052,"parameters":{"matrix":[[-11,-2],[0,-3]],"x":[-1,-2],"u":[1,-1],"v":[-2,1],"alpha":-1,"beta":-2,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-1,\"beta\":-2,\"matrix\":[[-11,-2],[0,-3]],\"u\":[1,-1],\"v\":[-2,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0053","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-10,-1],[5,-2]],\"x\":[0,-1],\"u\":[1,0],\"v\":[-1,1],\"alpha\":0,\"beta\":-1}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, -1, [1, -1], -7, -7, 是)","explanation_zh":"Wave4根節點核對=(0, -1, [1, -1], -7, -7, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274053,"parameters":{"matrix":[[-10,-1],[5,-2]],"x":[0,-1],"u":[1,0],"v":[-1,1],"alpha":0,"beta":-1,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":0,\"beta\":-1,\"matrix\":[[-10,-1],[5,-2]],\"u\":[1,0],\"v\":[-1,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0054","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-9,0],[-1,-1]],\"x\":[1,0],\"u\":[1,1],\"v\":[0,1],\"alpha\":1,\"beta\":0}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [1, 1], -9, -9, 是)","explanation_zh":"Wave4根節點核對=(1, 0, [1, 1], -9, -9, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274054,"parameters":{"matrix":[[-9,0],[-1,-1]],"x":[1,0],"u":[1,1],"v":[0,1],"alpha":1,"beta":0,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":1,\"beta\":0,\"matrix\":[[-9,0],[-1,-1]],\"u\":[1,1],\"v\":[0,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0055","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-8,1],[4,0]],\"x\":[2,1],\"u\":[1,2],\"v\":[1,1],\"alpha\":2,\"beta\":1}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, [3, 5], -26, -26, 是)","explanation_zh":"Wave4根節點核對=(2, 1, [3, 5], -26, -26, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274055,"parameters":{"matrix":[[-8,1],[4,0]],"x":[2,1],"u":[1,2],"v":[1,1],"alpha":2,"beta":1,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":2,\"beta\":1,\"matrix\":[[-8,1],[4,0]],\"u\":[1,2],\"v\":[1,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0056","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-7,2],[-2,1]],\"x\":[-2,2],\"u\":[1,-2],\"v\":[2,1],\"alpha\":-2,\"beta\":2}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, 2, [2, 6], 8, 8, 是)","explanation_zh":"Wave4根節點核對=(-2, 2, [2, 6], 8, 8, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274056,"parameters":{"matrix":[[-7,2],[-2,1]],"x":[-2,2],"u":[1,-2],"v":[2,1],"alpha":-2,"beta":2,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-2,\"beta\":2,\"matrix\":[[-7,2],[-2,1]],\"u\":[1,-2],\"v\":[2,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0057","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-6,3],[3,2]],\"x\":[-1,3],\"u\":[1,-1],\"v\":[3,1],\"alpha\":-1,\"beta\":3}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, 3, [8, 4], 132, 132, 是)","explanation_zh":"Wave4根節點核對=(-1, 3, [8, 4], 132, 132, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274057,"parameters":{"matrix":[[-6,3],[3,2]],"x":[-1,3],"u":[1,-1],"v":[3,1],"alpha":-1,"beta":3,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-1,\"beta\":3,\"matrix\":[[-6,3],[3,2]],\"u\":[1,-1],\"v\":[3,1],\"x\":[-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0058","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-5,-3],[-3,3]],\"x\":[0,-3],\"u\":[1,0],\"v\":[-3,1],\"alpha\":0,\"beta\":-3}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, -3, [9, -3], 108, 108, 是)","explanation_zh":"Wave4根節點核對=(0, -3, [9, -3], 108, 108, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274058,"parameters":{"matrix":[[-5,-3],[-3,3]],"x":[0,-3],"u":[1,0],"v":[-3,1],"alpha":0,"beta":-3,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":0,\"beta\":-3,\"matrix\":[[-5,-3],[-3,3]],\"u\":[1,0],\"v\":[-3,1],\"x\":[0,-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0059","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-4,-2],[2,4]],\"x\":[1,-2],\"u\":[1,1],\"v\":[-2,1],\"alpha\":1,\"beta\":-2}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, -2, [5, -1], -30, -30, 是)","explanation_zh":"Wave4根節點核對=(1, -2, [5, -1], -30, -30, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274059,"parameters":{"matrix":[[-4,-2],[2,4]],"x":[1,-2],"u":[1,1],"v":[-2,1],"alpha":1,"beta":-2,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":1,\"beta\":-2,\"matrix\":[[-4,-2],[2,4]],\"u\":[1,1],\"v\":[-2,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0060","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-3,-1],[-4,-4]],\"x\":[2,-1],\"u\":[1,2],\"v\":[-1,1],\"alpha\":2,\"beta\":-1}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, -1, [3, 3], 0, 0, 是)","explanation_zh":"Wave4根節點核對=(2, -1, [3, 3], 0, 0, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274060,"parameters":{"matrix":[[-3,-1],[-4,-4]],"x":[2,-1],"u":[1,2],"v":[-1,1],"alpha":2,"beta":-1,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":2,\"beta\":-1,\"matrix\":[[-3,-1],[-4,-4]],\"u\":[1,2],\"v\":[-1,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0061","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-2,0],[1,-3]],\"x\":[-2,0],\"u\":[1,-2],\"v\":[0,1],\"alpha\":-2,\"beta\":0}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, 0, [-2, 4], -8, -8, 是)","explanation_zh":"Wave4根節點核對=(-2, 0, [-2, 4], -8, -8, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274061,"parameters":{"matrix":[[-2,0],[1,-3]],"x":[-2,0],"u":[1,-2],"v":[0,1],"alpha":-2,"beta":0,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-2,\"beta\":0,\"matrix\":[[-2,0],[1,-3]],\"u\":[1,-2],\"v\":[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0062","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-1,1],[-5,-2]],\"x\":[-1,1],\"u\":[1,-1],\"v\":[1,1],\"alpha\":-1,\"beta\":1}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, 1, [0, 2], -6, -6, 是)","explanation_zh":"Wave4根節點核對=(-1, 1, [0, 2], -6, -6, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274062,"parameters":{"matrix":[[-1,1],[-5,-2]],"x":[-1,1],"u":[1,-1],"v":[1,1],"alpha":-1,"beta":1,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-1,\"beta\":1,\"matrix\":[[-1,1],[-5,-2]],\"u\":[1,-1],\"v\":[1,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0063","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[0,2],[0,-1]],\"x\":[0,2],\"u\":[1,0],\"v\":[2,1],\"alpha\":0,\"beta\":2}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 2, [4, 2], -4, -4, 是)","explanation_zh":"Wave4根節點核對=(0, 2, [4, 2], -4, -4, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274063,"parameters":{"matrix":[[0,2],[0,-1]],"x":[0,2],"u":[1,0],"v":[2,1],"alpha":0,"beta":2,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":0,\"beta\":2,\"matrix\":[[0,2],[0,-1]],\"u\":[1,0],\"v\":[2,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0064","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[1,3],[5,0]],\"x\":[1,3],\"u\":[1,1],\"v\":[3,1],\"alpha\":1,\"beta\":3}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, [10, 4], 172, 172, 是)","explanation_zh":"Wave4根節點核對=(1, 3, [10, 4], 172, 172, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274064,"parameters":{"matrix":[[1,3],[5,0]],"x":[1,3],"u":[1,1],"v":[3,1],"alpha":1,"beta":3,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":1,\"beta\":3,\"matrix\":[[1,3],[5,0]],\"u\":[1,1],\"v\":[3,1],\"x\":[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0065","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[2,-3],[-1,1]],\"x\":[2,-3],\"u\":[1,2],\"v\":[-3,1],\"alpha\":2,\"beta\":-3}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, -3, [11, 1], 68, 68, 是)","explanation_zh":"Wave4根節點核對=(2, -3, [11, 1], 68, 68, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274065,"parameters":{"matrix":[[2,-3],[-1,1]],"x":[2,-3],"u":[1,2],"v":[-3,1],"alpha":2,"beta":-3,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":2,\"beta\":-3,\"matrix\":[[2,-3],[-1,1]],\"u\":[1,2],\"v\":[-3,1],\"x\":[2,-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0066","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[3,-2],[4,2]],\"x\":[-2,-2],\"u\":[1,-2],\"v\":[-2,1],\"alpha\":-2,\"beta\":-2}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, -2, [2, 2], -28, -28, 是)","explanation_zh":"Wave4根節點核對=(-2, -2, [2, 2], -28, -28, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274066,"parameters":{"matrix":[[3,-2],[4,2]],"x":[-2,-2],"u":[1,-2],"v":[-2,1],"alpha":-2,"beta":-2,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-2,\"beta\":-2,\"matrix\":[[3,-2],[4,2]],\"u\":[1,-2],\"v\":[-2,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0067","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[4,-1],[-2,3]],\"x\":[-1,-1],\"u\":[1,-1],\"v\":[-1,1],\"alpha\":-1,\"beta\":-1}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, -1, [0, 0], 0, 0, 是)","explanation_zh":"Wave4根節點核對=(-1, -1, [0, 0], 0, 0, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274067,"parameters":{"matrix":[[4,-1],[-2,3]],"x":[-1,-1],"u":[1,-1],"v":[-1,1],"alpha":-1,"beta":-1,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-1,\"beta\":-1,\"matrix\":[[4,-1],[-2,3]],\"u\":[1,-1],\"v\":[-1,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0068","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[5,0],[3,4]],\"x\":[0,0],\"u\":[1,0],\"v\":[0,1],\"alpha\":0,\"beta\":0}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, [0, 0], 0, 0, 是)","explanation_zh":"Wave4根節點核對=(0, 0, [0, 0], 0, 0, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274068,"parameters":{"matrix":[[5,0],[3,4]],"x":[0,0],"u":[1,0],"v":[0,1],"alpha":0,"beta":0,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":0,\"beta\":0,\"matrix\":[[5,0],[3,4]],\"u\":[1,0],\"v\":[0,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0069","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[6,1],[-3,-4]],\"x\":[1,1],\"u\":[1,1],\"v\":[1,1],\"alpha\":1,\"beta\":1}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [2, 2], 0, 0, 是)","explanation_zh":"Wave4根節點核對=(1, 1, [2, 2], 0, 0, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274069,"parameters":{"matrix":[[6,1],[-3,-4]],"x":[1,1],"u":[1,1],"v":[1,1],"alpha":1,"beta":1,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":1,\"beta\":1,\"matrix\":[[6,1],[-3,-4]],\"u\":[1,1],\"v\":[1,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0070","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[7,2],[2,-3]],\"x\":[2,2],\"u\":[1,2],\"v\":[2,1],\"alpha\":2,\"beta\":2}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [6, 6], 96, 96, 是)","explanation_zh":"Wave4根節點核對=(2, 2, [6, 6], 96, 96, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274070,"parameters":{"matrix":[[7,2],[2,-3]],"x":[2,2],"u":[1,2],"v":[2,1],"alpha":2,"beta":2,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":2,\"beta\":2,\"matrix\":[[7,2],[2,-3]],\"u\":[1,2],\"v\":[2,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0071","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[8,3],[-4,-2]],\"x\":[-2,3],\"u\":[1,-2],\"v\":[3,1],\"alpha\":-2,\"beta\":3}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, 3, [7, 7], -280, -280, 是)","explanation_zh":"Wave4根節點核對=(-2, 3, [7, 7], -280, -280, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274071,"parameters":{"matrix":[[8,3],[-4,-2]],"x":[-2,3],"u":[1,-2],"v":[3,1],"alpha":-2,"beta":3,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-2,\"beta\":3,\"matrix\":[[8,3],[-4,-2]],\"u\":[1,-2],\"v\":[3,1],\"x\":[-2,3]}"}},"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0072","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[9,-3],[1,-1]],\"x\":[-1,-3],\"u\":[1,-1],\"v\":[-3,1],\"alpha\":-1,\"beta\":-3}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(-1, -3, [8, -2], -108, -108, 是)","explanation_zh":"Wave4根節點核對=(-1, -3, [8, -2], -108, -108, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274072,"parameters":{"matrix":[[9,-3],[1,-1]],"x":[-1,-3],"u":[1,-1],"v":[-3,1],"alpha":-1,"beta":-3,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-1,\"beta\":-3,\"matrix\":[[9,-3],[1,-1]],\"u\":[1,-1],\"v\":[-3,1],\"x\":[-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0073","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[10,-2],[-5,0]],\"x\":[0,-2],\"u\":[1,0],\"v\":[-2,1],\"alpha\":0,\"beta\":-2}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, -2, [4, -2], 40, 40, 是)","explanation_zh":"Wave4根節點核對=(0, -2, [4, -2], 40, 40, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274073,"parameters":{"matrix":[[10,-2],[-5,0]],"x":[0,-2],"u":[1,0],"v":[-2,1],"alpha":0,"beta":-2,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":0,\"beta\":-2,\"matrix\":[[10,-2],[-5,0]],\"u\":[1,0],\"v\":[-2,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0074","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[11,-1],[0,1]],\"x\":[1,-1],\"u\":[1,1],\"v\":[-1,1],\"alpha\":1,\"beta\":-1}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, -1, [2, 0], 22, 22, 是)","explanation_zh":"Wave4根節點核對=(1, -1, [2, 0], 22, 22, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274074,"parameters":{"matrix":[[11,-1],[0,1]],"x":[1,-1],"u":[1,1],"v":[-1,1],"alpha":1,"beta":-1,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":1,\"beta\":-1,\"matrix\":[[11,-1],[0,1]],\"u\":[1,1],\"v\":[-1,1],\"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0075","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[12,0],[5,2]],\"x\":[2,0],\"u\":[1,2],\"v\":[0,1],\"alpha\":2,\"beta\":0}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [2, 4], 48, 48, 是)","explanation_zh":"Wave4根節點核對=(2, 0, [2, 4], 48, 48, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274075,"parameters":{"matrix":[[12,0],[5,2]],"x":[2,0],"u":[1,2],"v":[0,1],"alpha":2,"beta":0,"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-second-slot-identity","semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":2,\"beta\":0,\"matrix\":[[12,0],[5,2]],\"u\":[1,2],\"v\":[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0076","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[-12,-3],[-3,-4]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-12, -3], [-3, -4]], null, 是, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[-12, -3], [-3, -4]], null, 是, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274076,"parameters":{"matrix":[[-12,-3],[-3,-4]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-12,-3],[-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0077","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[-11,-2],[0,-3]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-11, -2], [0, -3]], [0, 1, -2, 0], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[-11, -2], [0, -3]], [0, 1, -2, 0], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274077,"parameters":{"matrix":[[-11,-2],[0,-3]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-11,-2],[0,-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0078","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[-10,-1],[5,-2]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-10, -1], [5, -2]], [0, 1, -1, 5], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[-10, -1], [5, -2]], [0, 1, -1, 5], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274078,"parameters":{"matrix":[[-10,-1],[5,-2]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-10,-1],[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0079","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[-9,0],[-1,-1]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-9, 0], [-1, -1]], [0, 1, 0, -1], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[-9, 0], [-1, -1]], [0, 1, 0, -1], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274079,"parameters":{"matrix":[[-9,0],[-1,-1]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-9,0],[-1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0080","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[-8,1],[1,0]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-8, 1], [1, 0]], null, 是, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[-8, 1], [1, 0]], null, 是, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274080,"parameters":{"matrix":[[-8,1],[1,0]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-8,1],[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0081","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[-7,2],[-2,1]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-7, 2], [-2, 1]], [0, 1, 2, -2], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[-7, 2], [-2, 1]], [0, 1, 2, -2], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274081,"parameters":{"matrix":[[-7,2],[-2,1]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-7,2],[-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0082","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[-6,3],[3,2]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-6, 3], [3, 2]], null, 是, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[-6, 3], [3, 2]], null, 是, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274082,"parameters":{"matrix":[[-6,3],[3,2]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-6,3],[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0083","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[-5,-3],[-3,3]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-5, -3], [-3, 3]], null, 是, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[-5, -3], [-3, 3]], null, 是, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274083,"parameters":{"matrix":[[-5,-3],[-3,3]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-5,-3],[-3,3]]}"}},"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0084","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[-4,-2],[-2,4]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-4, -2], [-2, 4]], null, 是, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[-4, -2], [-2, 4]], null, 是, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274084,"parameters":{"matrix":[[-4,-2],[-2,4]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-4,-2],[-2,4]]}"}},"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0085","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[-3,-1],[-4,-4]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-3, -1], [-4, -4]], [0, 1, -1, -4], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[-3, -1], [-4, -4]], [0, 1, -1, -4], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274085,"parameters":{"matrix":[[-3,-1],[-4,-4]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-3,-1],[-4,-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0086","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[-2,0],[1,-3]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-2, 0], [1, -3]], [0, 1, 0, 1], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[-2, 0], [1, -3]], [0, 1, 0, 1], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274086,"parameters":{"matrix":[[-2,0],[1,-3]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-2,0],[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0087","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[-1,1],[-5,-2]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-1, 1], [-5, -2]], [0, 1, 1, -5], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[-1, 1], [-5, -2]], [0, 1, 1, -5], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274087,"parameters":{"matrix":[[-1,1],[-5,-2]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-1,1],[-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0088","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[0,2],[2,-1]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2], [2, -1]], null, 是, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[0, 2], [2, -1]], null, 是, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274088,"parameters":{"matrix":[[0,2],[2,-1]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[0,2],[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0089","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[1,3],[5,0]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [5, 0]], [0, 1, 3, 5], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[1, 3], [5, 0]], [0, 1, 3, 5], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274089,"parameters":{"matrix":[[1,3],[5,0]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[1,3],[5,0]]}"}},"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0090","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[2,-3],[-1,1]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[2, -3], [-1, 1]], [0, 1, -3, -1], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[2, -3], [-1, 1]], [0, 1, -3, -1], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274090,"parameters":{"matrix":[[2,-3],[-1,1]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[2,-3],[-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0091","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[3,-2],[4,2]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[3, -2], [4, 2]], [0, 1, -2, 4], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[3, -2], [4, 2]], [0, 1, -2, 4], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274091,"parameters":{"matrix":[[3,-2],[4,2]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[3,-2],[4,2]]}"}},"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0092","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[4,-1],[-1,3]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[4, -1], [-1, 3]], null, 是, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[4, -1], [-1, 3]], null, 是, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274092,"parameters":{"matrix":[[4,-1],[-1,3]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[4,-1],[-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0093","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[5,0],[3,4]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[5, 0], [3, 4]], [0, 1, 0, 3], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[5, 0], [3, 4]], [0, 1, 0, 3], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274093,"parameters":{"matrix":[[5,0],[3,4]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[5,0],[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0094","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[6,1],[-3,-4]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[6, 1], [-3, -4]], [0, 1, 1, -3], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[6, 1], [-3, -4]], [0, 1, 1, -3], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274094,"parameters":{"matrix":[[6,1],[-3,-4]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[6,1],[-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0095","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[7,2],[2,-3]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[7, 2], [2, -3]], null, 是, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[7, 2], [2, -3]], null, 是, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274095,"parameters":{"matrix":[[7,2],[2,-3]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[7,2],[2,-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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0096","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[8,3],[3,-2]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[8, 3], [3, -2]], null, 是, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[8, 3], [3, -2]], null, 是, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274096,"parameters":{"matrix":[[8,3],[3,-2]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[8,3],[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0097","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[9,-3],[1,-1]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[9, -3], [1, -1]], [0, 1, -3, 1], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[9, -3], [1, -1]], [0, 1, -3, 1], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274097,"parameters":{"matrix":[[9,-3],[1,-1]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[9,-3],[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0098","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[10,-2],[-5,0]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[10, -2], [-5, 0]], [0, 1, -2, -5], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[10, -2], [-5, 0]], [0, 1, -2, -5], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274098,"parameters":{"matrix":[[10,-2],[-5,0]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[10,-2],[-5,0]]}"}},"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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0099","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[11,-1],[0,1]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[11, -1], [0, 1]], [0, 1, -1, 0], 否, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[11, -1], [0, 1]], [0, 1, -1, 0], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274099,"parameters":{"matrix":[[11,-1],[0,1]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[11,-1],[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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r04-real-bilinear-form-0100","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","language":"zh-Hant","source_mko_ids":["mko-real-bilinear-form"],"title_zh":"實雙線性形式","stem_zh":"雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[12,0],[0,2]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[12, 0], [0, 2]], null, 是, symmetry_is_extra_not_bilinearity)","explanation_zh":"Wave4根節點核對=([[12, 0], [0, 2]], null, 是, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實雙線性形式"]},"variant":{"family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274100,"parameters":{"matrix":[[12,0],[0,2]],"domain":"Exact_matrix_real_bilinear_form","operation":"bilinear-symmetry-audit","semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[12,0],[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、一般完備性、內積／有界性、mutual independence、directed graph、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"}}