{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0001","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-12,-3,-2]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-12, -3, -2], [-14, -5])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-12, -3, -2], [-14, -5])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232001,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-12,-3,-2],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-12,-3,-2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0002","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-11,-2,-1]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-11, -2, -1], [-12, -3])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-11, -2, -1], [-12, -3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232002,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-11,-2,-1],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-11,-2,-1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0003","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-10,-1,0]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-10, -1, 0], [-10, -1])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-10, -1, 0], [-10, -1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232003,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-10,-1,0],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-10,-1,0],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0004","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-9,0,1]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-9, 0, 1], [-8, 1])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-9, 0, 1], [-8, 1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232004,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-9,0,1],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-9,0,1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0005","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-8,1,2]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-8, 1, 2], [-6, 3])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-8, 1, 2], [-6, 3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232005,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-8,1,2],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-8,1,2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0006","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-7,2,-2]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-7, 2, -2], [-9, 0])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-7, 2, -2], [-9, 0])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232006,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-7,2,-2],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-7,2,-2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0007","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-6,3,-1]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-6, 3, -1], [-7, 2])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-6, 3, -1], [-7, 2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232007,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-6,3,-1],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-6,3,-1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0008","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-5,-3,0]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-5, -3, 0], [-5, -3])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-5, -3, 0], [-5, -3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232008,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-5,-3,0],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-5,-3,0],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0009","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-4,-2,1]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-3, -1])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-3, -1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232009,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-4,-2,1],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-4,-2,1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0010","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-3,-1,2]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-3, -1, 2], [-1, 1])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-3, -1, 2], [-1, 1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232010,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-3,-1,2],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-3,-1,2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0011","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-2,0,-2]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-2, 0, -2], [-4, -2])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-2, 0, -2], [-4, -2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232011,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-2,0,-2],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-2,0,-2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0012","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-1,1,-1]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-1, 1, -1], [-2, 0])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-1, 1, -1], [-2, 0])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232012,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-1,1,-1],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-1,1,-1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0013","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[0,2,0]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [0, 2, 0], [0, 2])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [0, 2, 0], [0, 2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232013,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[0,2,0],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[0,2,0],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0014","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[1,3,1]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [1, 3, 1], [2, 4])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [1, 3, 1], [2, 4])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232014,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[1,3,1],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[1,3,1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0015","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[2,-3,2]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [2, -3, 2], [4, -1])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [2, -3, 2], [4, -1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232015,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[2,-3,2],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[2,-3,2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0016","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[3,-2,-2]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [3, -2, -2], [1, -4])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [3, -2, -2], [1, -4])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232016,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[3,-2,-2],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[3,-2,-2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0017","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[4,-1,-1]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [4, -1, -1], [3, -2])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [4, -1, -1], [3, -2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232017,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[4,-1,-1],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[4,-1,-1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0018","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[5,0,0]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [5, 0, 0], [5, 0])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [5, 0, 0], [5, 0])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232018,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[5,0,0],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[5,0,0],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0019","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[6,1,1]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [6, 1, 1], [7, 2])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [6, 1, 1], [7, 2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232019,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[6,1,1],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[6,1,1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0020","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[7,2,2]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [7, 2, 2], [9, 4])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [7, 2, 2], [9, 4])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232020,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[7,2,2],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[7,2,2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0021","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[8,3,-2]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [8, 3, -2], [6, 1])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [8, 3, -2], [6, 1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232021,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[8,3,-2],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[8,3,-2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0022","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[9,-3,-1]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [9, -3, -1], [8, -4])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [9, -3, -1], [8, -4])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232022,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[9,-3,-1],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[9,-3,-1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0023","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[10,-2,0]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [10, -2, 0], [10, -2])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [10, -2, 0], [10, -2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232023,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[10,-2,0],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[10,-2,0],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0024","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[11,-1,1]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [11, -1, 1], [12, 0])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [11, -1, 1], [12, 0])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232024,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[11,-1,1],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[11,-1,1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0025","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[12,0,2]}。請依 [vectors,coefficients,sum] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [12, 0, 2], [14, 2])","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [12, 0, 2], [14, 2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-weighted-sum","seed":202610232025,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[12,0,2],"domain":"Finite_coordinate_linear_combinations","operation":"span-weighted-sum","semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[12,0,2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0026","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-12,-3,-2],\"target\":[-13,-5]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-12, -3, -2], [-13, -5], [-14, -5], 否)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-12, -3, -2], [-13, -5], [-14, -5], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232026,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-12,-3,-2],"target":[-13,-5],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-12,-3,-2],\"target\":[-13,-5],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0027","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-11,-2,-1],\"target\":[-12,-3]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-11, -2, -1], [-12, -3], [-12, -3], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-11, -2, -1], [-12, -3], [-12, -3], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232027,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-11,-2,-1],"target":[-12,-3],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-11,-2,-1],\"target\":[-12,-3],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0028","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-10,-1,0],\"target\":[-10,-1]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-10, -1, 0], [-10, -1], [-10, -1], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-10, -1, 0], [-10, -1], [-10, -1], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232028,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-10,-1,0],"target":[-10,-1],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-10,-1,0],\"target\":[-10,-1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0029","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-9,0,1],\"target\":[-8,1]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-9, 0, 1], [-8, 1], [-8, 1], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-9, 0, 1], [-8, 1], [-8, 1], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232029,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-9,0,1],"target":[-8,1],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-9,0,1],\"target\":[-8,1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0030","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-8,1,2],\"target\":[-5,3]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-8, 1, 2], [-5, 3], [-6, 3], 否)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-8, 1, 2], [-5, 3], [-6, 3], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232030,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-8,1,2],"target":[-5,3],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-8,1,2],\"target\":[-5,3],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0031","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-7,2,-2],\"target\":[-9,0]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-7, 2, -2], [-9, 0], [-9, 0], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-7, 2, -2], [-9, 0], [-9, 0], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232031,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-7,2,-2],"target":[-9,0],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-7,2,-2],\"target\":[-9,0],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0032","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-6,3,-1],\"target\":[-7,2]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-6, 3, -1], [-7, 2], [-7, 2], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-6, 3, -1], [-7, 2], [-7, 2], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232032,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-6,3,-1],"target":[-7,2],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-6,3,-1],\"target\":[-7,2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0033","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-5,-3,0],\"target\":[-5,-3]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-5, -3, 0], [-5, -3], [-5, -3], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-5, -3, 0], [-5, -3], [-5, -3], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232033,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-5,-3,0],"target":[-5,-3],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-5,-3,0],\"target\":[-5,-3],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0034","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-4,-2,1],\"target\":[-2,-1]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-2, -1], [-3, -1], 否)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-2, -1], [-3, -1], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232034,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-4,-2,1],"target":[-2,-1],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-4,-2,1],\"target\":[-2,-1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0035","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-3,-1,2],\"target\":[-1,1]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-3, -1, 2], [-1, 1], [-1, 1], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-3, -1, 2], [-1, 1], [-1, 1], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232035,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-3,-1,2],"target":[-1,1],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-3,-1,2],\"target\":[-1,1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0036","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-2,0,-2],\"target\":[-4,-2]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-2, 0, -2], [-4, -2], [-4, -2], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-2, 0, -2], [-4, -2], [-4, -2], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232036,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-2,0,-2],"target":[-4,-2],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-2,0,-2],\"target\":[-4,-2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0037","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-1,1,-1],\"target\":[-2,0]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-1, 1, -1], [-2, 0], [-2, 0], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-1, 1, -1], [-2, 0], [-2, 0], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232037,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-1,1,-1],"target":[-2,0],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-1,1,-1],\"target\":[-2,0],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0038","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[0,2,0],\"target\":[1,2]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [0, 2, 0], [1, 2], [0, 2], 否)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [0, 2, 0], [1, 2], [0, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232038,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[0,2,0],"target":[1,2],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[0,2,0],\"target\":[1,2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0039","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[1,3,1],\"target\":[2,4]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [1, 3, 1], [2, 4], [2, 4], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [1, 3, 1], [2, 4], [2, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232039,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[1,3,1],"target":[2,4],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[1,3,1],\"target\":[2,4],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0040","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[2,-3,2],\"target\":[4,-1]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [2, -3, 2], [4, -1], [4, -1], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [2, -3, 2], [4, -1], [4, -1], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232040,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[2,-3,2],"target":[4,-1],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[2,-3,2],\"target\":[4,-1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0041","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[3,-2,-2],\"target\":[1,-4]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [3, -2, -2], [1, -4], [1, -4], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [3, -2, -2], [1, -4], [1, -4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232041,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[3,-2,-2],"target":[1,-4],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[3,-2,-2],\"target\":[1,-4],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0042","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[4,-1,-1],\"target\":[4,-2]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [4, -1, -1], [4, -2], [3, -2], 否)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [4, -1, -1], [4, -2], [3, -2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232042,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[4,-1,-1],"target":[4,-2],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[4,-1,-1],\"target\":[4,-2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0043","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[5,0,0],\"target\":[5,0]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [5, 0, 0], [5, 0], [5, 0], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [5, 0, 0], [5, 0], [5, 0], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232043,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[5,0,0],"target":[5,0],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[5,0,0],\"target\":[5,0],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0044","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[6,1,1],\"target\":[7,2]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [6, 1, 1], [7, 2], [7, 2], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [6, 1, 1], [7, 2], [7, 2], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232044,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[6,1,1],"target":[7,2],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[6,1,1],\"target\":[7,2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0045","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[7,2,2],\"target\":[9,4]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [7, 2, 2], [9, 4], [9, 4], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [7, 2, 2], [9, 4], [9, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232045,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[7,2,2],"target":[9,4],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[7,2,2],\"target\":[9,4],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0046","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[8,3,-2],\"target\":[7,1]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [8, 3, -2], [7, 1], [6, 1], 否)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [8, 3, -2], [7, 1], [6, 1], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232046,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[8,3,-2],"target":[7,1],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[8,3,-2],\"target\":[7,1],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0047","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[9,-3,-1],\"target\":[8,-4]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [9, -3, -1], [8, -4], [8, -4], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [9, -3, -1], [8, -4], [8, -4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232047,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[9,-3,-1],"target":[8,-4],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[9,-3,-1],\"target\":[8,-4],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0048","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[10,-2,0],\"target\":[10,-2]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [10, -2, 0], [10, -2], [10, -2], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [10, -2, 0], [10, -2], [10, -2], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232048,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[10,-2,0],"target":[10,-2],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[10,-2,0],\"target\":[10,-2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0049","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[11,-1,1],\"target\":[12,0]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [11, -1, 1], [12, 0], [12, 0], 是)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [11, -1, 1], [12, 0], [12, 0], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232049,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[11,-1,1],"target":[12,0],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[11,-1,1],\"target\":[12,0],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0050","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[12,0,2],\"target\":[15,2]}。請依 [vectors,coefficients,target,sum,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [12, 0, 2], [15, 2], [14, 2], 否)","explanation_zh":"現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [12, 0, 2], [15, 2], [14, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-witness-audit","seed":202610232050,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[12,0,2],"target":[15,2],"domain":"Finite_coordinate_linear_combinations","operation":"span-witness-audit","semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[12,0,2],\"target\":[15,2],\"vectors\":[[0,1],[1,0],[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0051","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[-12,-3]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-12, -3], -2, -15/2, -9/2, [-12, -3])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [-12, -3], -2, -15/2, -9/2, [-12, -3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232051,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[-12,-3],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-12,-3],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0052","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[-11,-2]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-11, -2], -2, -13/2, -9/2, [-11, -2])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [-11, -2], -2, -13/2, -9/2, [-11, -2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232052,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[-11,-2],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-11,-2],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0053","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[-10,-1]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-10, -1], -2, -11/2, -9/2, [-10, -1])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [-10, -1], -2, -11/2, -9/2, [-10, -1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232053,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[-10,-1],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-10,-1],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0054","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[-9,0]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-9, 0], -2, -9/2, -9/2, [-9, 0])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [-9, 0], -2, -9/2, -9/2, [-9, 0])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232054,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[-9,0],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-9,0],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0055","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[-8,1]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-8, 1], -2, -7/2, -9/2, [-8, 1])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [-8, 1], -2, -7/2, -9/2, [-8, 1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232055,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[-8,1],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-8,1],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0056","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[-7,2]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-7, 2], -2, -5/2, -9/2, [-7, 2])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [-7, 2], -2, -5/2, -9/2, [-7, 2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232056,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[-7,2],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-7,2],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0057","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[-6,3]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-6, 3], -2, -3/2, -9/2, [-6, 3])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [-6, 3], -2, -3/2, -9/2, [-6, 3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232057,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[-6,3],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-6,3],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0058","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[-5,-3]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-5, -3], -2, -4, -1, [-5, -3])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [-5, -3], -2, -4, -1, [-5, -3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232058,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[-5,-3],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-5,-3],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0059","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[-4,-2]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-4, -2], -2, -3, -1, [-4, -2])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [-4, -2], -2, -3, -1, [-4, -2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232059,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[-4,-2],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-4,-2],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0060","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[-3,-1]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-3, -1], -2, -2, -1, [-3, -1])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [-3, -1], -2, -2, -1, [-3, -1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232060,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[-3,-1],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-3,-1],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0061","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[-2,0]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-2, 0], -2, -1, -1, [-2, 0])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [-2, 0], -2, -1, -1, [-2, 0])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232061,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[-2,0],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-2,0],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0062","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[-1,1]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-1, 1], -2, 0, -1, [-1, 1])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [-1, 1], -2, 0, -1, [-1, 1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232062,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[-1,1],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-1,1],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0063","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[0,2]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [0, 2], -2, 1, -1, [0, 2])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [0, 2], -2, 1, -1, [0, 2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232063,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[0,2],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[0,2],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0064","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[1,3]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [1, 3], -2, 2, -1, [1, 3])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [1, 3], -2, 2, -1, [1, 3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232064,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[1,3],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[1,3],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0065","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[2,-3]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [2, -3], -2, -1/2, 5/2, [2, -3])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [2, -3], -2, -1/2, 5/2, [2, -3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232065,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[2,-3],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[2,-3],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0066","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[3,-2]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [3, -2], -2, 1/2, 5/2, [3, -2])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [3, -2], -2, 1/2, 5/2, [3, -2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232066,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[3,-2],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[3,-2],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0067","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[4,-1]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [4, -1], -2, 3/2, 5/2, [4, -1])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [4, -1], -2, 3/2, 5/2, [4, -1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232067,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[4,-1],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[4,-1],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0068","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[5,0]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [5, 0], -2, 5/2, 5/2, [5, 0])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [5, 0], -2, 5/2, 5/2, [5, 0])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232068,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[5,0],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[5,0],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0069","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[6,1]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [6, 1], -2, 7/2, 5/2, [6, 1])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [6, 1], -2, 7/2, 5/2, [6, 1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232069,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[6,1],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[6,1],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0070","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[7,2]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [7, 2], -2, 9/2, 5/2, [7, 2])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [7, 2], -2, 9/2, 5/2, [7, 2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232070,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[7,2],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[7,2],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0071","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[8,3]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [8, 3], -2, 11/2, 5/2, [8, 3])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [8, 3], -2, 11/2, 5/2, [8, 3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232071,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[8,3],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[8,3],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0072","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[9,-3]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [9, -3], -2, 3, 6, [9, -3])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [9, -3], -2, 3, 6, [9, -3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232072,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[9,-3],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[9,-3],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0073","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[10,-2]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [10, -2], -2, 4, 6, [10, -2])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [10, -2], -2, 4, 6, [10, -2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232073,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[10,-2],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[10,-2],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0074","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[11,-1]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [11, -1], -2, 5, 6, [11, -1])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [11, -1], -2, 5, 6, [11, -1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232074,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[11,-1],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[11,-1],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0075","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[12,0]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -1], [12, 0], -2, 6, 6, [12, 0])","explanation_zh":"現代橋梁核對=([1, 1], [1, -1], [12, 0], -2, 6, 6, [12, 0])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232075,"parameters":{"v1":[1,1],"v2":[1,-1],"target":[12,0],"domain":"Finite_coordinate_linear_combinations","operation":"span-r2-coordinate-solve","semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[12,0],\"v1\":[1,1],\"v2\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0076","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[-20,-39]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [-20, -39], -10, 否)","explanation_zh":"現代橋梁核對=([2, 4], [-20, -39], -10, 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232076,"parameters":{"generator":[2,4],"target":[-20,-39],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-20,-39]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0077","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[-18,-36]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [-18, -36], -9, 是)","explanation_zh":"現代橋梁核對=([2, 4], [-18, -36], -9, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232077,"parameters":{"generator":[2,4],"target":[-18,-36],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-18,-36]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0078","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[-16,-32]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [-16, -32], -8, 是)","explanation_zh":"現代橋梁核對=([2, 4], [-16, -32], -8, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232078,"parameters":{"generator":[2,4],"target":[-16,-32],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-16,-32]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0079","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[-14,-28]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [-14, -28], -7, 是)","explanation_zh":"現代橋梁核對=([2, 4], [-14, -28], -7, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232079,"parameters":{"generator":[2,4],"target":[-14,-28],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-14,-28]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0080","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[-12,-23]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [-12, -23], -6, 否)","explanation_zh":"現代橋梁核對=([2, 4], [-12, -23], -6, 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232080,"parameters":{"generator":[2,4],"target":[-12,-23],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-12,-23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0081","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[-10,-20]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [-10, -20], -5, 是)","explanation_zh":"現代橋梁核對=([2, 4], [-10, -20], -5, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232081,"parameters":{"generator":[2,4],"target":[-10,-20],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-10,-20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0082","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[-8,-16]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [-8, -16], -4, 是)","explanation_zh":"現代橋梁核對=([2, 4], [-8, -16], -4, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232082,"parameters":{"generator":[2,4],"target":[-8,-16],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-8,-16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0083","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[-6,-12]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [-6, -12], -3, 是)","explanation_zh":"現代橋梁核對=([2, 4], [-6, -12], -3, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232083,"parameters":{"generator":[2,4],"target":[-6,-12],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-6,-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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0084","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[-4,-7]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [-4, -7], -2, 否)","explanation_zh":"現代橋梁核對=([2, 4], [-4, -7], -2, 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232084,"parameters":{"generator":[2,4],"target":[-4,-7],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-4,-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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0085","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[-2,-4]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [-2, -4], -1, 是)","explanation_zh":"現代橋梁核對=([2, 4], [-2, -4], -1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232085,"parameters":{"generator":[2,4],"target":[-2,-4],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0087","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[2,4]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [2, 4], 1, 是)","explanation_zh":"現代橋梁核對=([2, 4], [2, 4], 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232087,"parameters":{"generator":[2,4],"target":[2,4],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0088","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[4,9]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [4, 9], 2, 否)","explanation_zh":"現代橋梁核對=([2, 4], [4, 9], 2, 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232088,"parameters":{"generator":[2,4],"target":[4,9],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[4,9]}"}},"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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0089","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[6,12]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [6, 12], 3, 是)","explanation_zh":"現代橋梁核對=([2, 4], [6, 12], 3, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232089,"parameters":{"generator":[2,4],"target":[6,12],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[6,12]}"}},"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、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0090","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[8,16]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [8, 16], 4, 是)","explanation_zh":"現代橋梁核對=([2, 4], [8, 16], 4, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232090,"parameters":{"generator":[2,4],"target":[8,16],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[8,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0091","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[10,20]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [10, 20], 5, 是)","explanation_zh":"現代橋梁核對=([2, 4], [10, 20], 5, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232091,"parameters":{"generator":[2,4],"target":[10,20],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[10,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0092","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[12,25]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [12, 25], 6, 否)","explanation_zh":"現代橋梁核對=([2, 4], [12, 25], 6, 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232092,"parameters":{"generator":[2,4],"target":[12,25],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[12,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0093","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[14,28]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [14, 28], 7, 是)","explanation_zh":"現代橋梁核對=([2, 4], [14, 28], 7, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232093,"parameters":{"generator":[2,4],"target":[14,28],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[14,28]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0094","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[16,32]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [16, 32], 8, 是)","explanation_zh":"現代橋梁核對=([2, 4], [16, 32], 8, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232094,"parameters":{"generator":[2,4],"target":[16,32],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[16,32]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0095","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[18,36]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [18, 36], 9, 是)","explanation_zh":"現代橋梁核對=([2, 4], [18, 36], 9, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232095,"parameters":{"generator":[2,4],"target":[18,36],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[18,36]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0096","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[20,41]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [20, 41], 10, 否)","explanation_zh":"現代橋梁核對=([2, 4], [20, 41], 10, 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232096,"parameters":{"generator":[2,4],"target":[20,41],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[20,41]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0097","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[22,44]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [22, 44], 11, 是)","explanation_zh":"現代橋梁核對=([2, 4], [22, 44], 11, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232097,"parameters":{"generator":[2,4],"target":[22,44],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[22,44]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0098","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[24,48]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [24, 48], 12, 是)","explanation_zh":"現代橋梁核對=([2, 4], [24, 48], 12, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232098,"parameters":{"generator":[2,4],"target":[24,48],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[24,48]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0099","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[26,52]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [26, 52], 13, 是)","explanation_zh":"現代橋梁核對=([2, 4], [26, 52], 13, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232099,"parameters":{"generator":[2,4],"target":[26,52],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[26,52]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r02-linear-combination-span-0100","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-combination-span"],"title_zh":"有限線性組合","stem_zh":"固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[28,57]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [28, 57], 14, 否)","explanation_zh":"現代橋梁核對=([2, 4], [28, 57], 14, 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限線性組合"]},"variant":{"family_id":"qf-mod23-span-single-generator-membership","seed":202610232100,"parameters":{"generator":[2,4],"target":[28,57],"domain":"Finite_coordinate_linear_combinations","operation":"span-single-generator-membership","semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[28,57]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}