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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r04-linear-independence-basis-0002","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[-11,-2],\"target\":[-11,-2]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-11, -2], [-11, -2], [-11, -2], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [-11, -2], [-11, -2], [-11, -2], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254002,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[-11,-2],"target":[-11,-2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-11,-2],\"target\":[-11,-2],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0003","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[-10,-1],\"target\":[-10,-1]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-10, -1], [-10, -1], [-10, -1], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [-10, -1], [-10, -1], [-10, -1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254003,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[-10,-1],"target":[-10,-1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-10,-1],\"target\":[-10,-1],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0004","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[-9,0],\"target\":[-9,0]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-9, 0], [-9, 0], [-9, 0], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [-9, 0], [-9, 0], [-9, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254004,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[-9,0],"target":[-9,0],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-9,0],\"target\":[-9,0],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0005","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[-8,1],\"target\":[-8,1]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-8, 1], [-8, 1], [-8, 1], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [-8, 1], [-8, 1], [-8, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254005,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[-8,1],"target":[-8,1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-8,1],\"target\":[-8,1],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0006","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[-7,2],\"target\":[-6,2]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-7, 2], [-6, 2], [-7, 2], 否)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [-7, 2], [-6, 2], [-7, 2], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254006,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[-7,2],"target":[-6,2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-7,2],\"target\":[-6,2],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0007","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[-6,3],\"target\":[-6,3]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-6, 3], [-6, 3], [-6, 3], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [-6, 3], [-6, 3], [-6, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254007,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[-6,3],"target":[-6,3],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-6,3],\"target\":[-6,3],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0008","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[-5,-3],\"target\":[-5,-3]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-5, -3], [-5, -3], [-5, -3], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [-5, -3], [-5, -3], [-5, -3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254008,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[-5,-3],"target":[-5,-3],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-5,-3],\"target\":[-5,-3],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0009","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[-4,-2],\"target\":[-4,-2]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-4, -2], [-4, -2], [-4, -2], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [-4, -2], [-4, -2], [-4, -2], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254009,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[-4,-2],"target":[-4,-2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-4,-2],\"target\":[-4,-2],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0010","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[-3,-1],\"target\":[-3,-1]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-3, -1], [-3, -1], [-3, -1], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [-3, -1], [-3, -1], [-3, -1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254010,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[-3,-1],"target":[-3,-1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-3,-1],\"target\":[-3,-1],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0011","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[-2,0],\"target\":[-1,0]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-2, 0], [-1, 0], [-2, 0], 否)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [-2, 0], [-1, 0], [-2, 0], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254011,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[-2,0],"target":[-1,0],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-2,0],\"target\":[-1,0],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0012","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[-1,1],\"target\":[-1,1]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-1, 1], [-1, 1], [-1, 1], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [-1, 1], [-1, 1], [-1, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254012,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[-1,1],"target":[-1,1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-1,1],\"target\":[-1,1],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0013","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[0,2],\"target\":[0,2]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [0, 2], [0, 2], [0, 2], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [0, 2], [0, 2], [0, 2], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254013,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[0,2],"target":[0,2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[0,2],\"target\":[0,2],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0014","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[1,3],\"target\":[1,3]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [1, 3], [1, 3], [1, 3], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [1, 3], [1, 3], [1, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254014,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[1,3],"target":[1,3],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[1,3],\"target\":[1,3],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0015","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[2,-3],\"target\":[2,-3]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [2, -3], [2, -3], [2, -3], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [2, -3], [2, -3], [2, -3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254015,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[2,-3],"target":[2,-3],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[2,-3],\"target\":[2,-3],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0016","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[3,-2],\"target\":[4,-2]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [3, -2], [4, -2], [3, -2], 否)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [3, -2], [4, -2], [3, -2], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254016,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[3,-2],"target":[4,-2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[3,-2],\"target\":[4,-2],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0017","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[4,-1],\"target\":[4,-1]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [4, -1], [4, -1], [4, -1], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [4, -1], [4, -1], [4, -1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254017,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[4,-1],"target":[4,-1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[4,-1],\"target\":[4,-1],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0018","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[5,0],\"target\":[5,0]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [5, 0], [5, 0], [5, 0], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [5, 0], [5, 0], [5, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254018,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[5,0],"target":[5,0],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[5,0],\"target\":[5,0],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0019","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[6,1],\"target\":[6,1]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [6, 1], [6, 1], [6, 1], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [6, 1], [6, 1], [6, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254019,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[6,1],"target":[6,1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[6,1],\"target\":[6,1],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0020","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[7,2],\"target\":[7,2]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [7, 2], [7, 2], [7, 2], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [7, 2], [7, 2], [7, 2], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254020,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[7,2],"target":[7,2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[7,2],\"target\":[7,2],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0021","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[8,3],\"target\":[9,3]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [8, 3], [9, 3], [8, 3], 否)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [8, 3], [9, 3], [8, 3], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254021,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[8,3],"target":[9,3],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[8,3],\"target\":[9,3],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0022","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[9,-3],\"target\":[9,-3]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [9, -3], [9, -3], [9, -3], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [9, -3], [9, -3], [9, -3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254022,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[9,-3],"target":[9,-3],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[9,-3],\"target\":[9,-3],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0023","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[10,-2],\"target\":[10,-2]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [10, -2], [10, -2], [10, -2], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [10, -2], [10, -2], [10, -2], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254023,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[10,-2],"target":[10,-2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[10,-2],\"target\":[10,-2],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0024","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-span-witness-audit；資料={\"vectors\":[[1,0],[0,1]],\"coefficients\":[11,-1],\"target\":[11,-1]}。請依 [vectors,coefficients,target,reconstructed,matches] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [11, -1], [11, -1], [11, -1], 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [11, -1], [11, -1], [11, -1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254024,"parameters":{"vectors":[[1,0],[0,1]],"coefficients":[11,-1],"target":[11,-1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-span-witness-audit","semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[11,-1],\"target\":[11,-1],\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0027","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,-11],[0,0],[1,1]],\"zero_index\":1}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, -11], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)","explanation_zh":"收束波核對=([[1, -11], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254027,"parameters":{"vectors":[[1,-11],[0,0],[1,1]],"zero_index":1,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-11],[0,0],[1,1]],\"zero_index\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0028","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,-10],[0,1],[0,0]],\"zero_index\":2}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, -10], [0, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)","explanation_zh":"收束波核對=([[1, -10], [0, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254028,"parameters":{"vectors":[[1,-10],[0,1],[0,0]],"zero_index":2,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-10],[0,1],[0,0]],\"zero_index\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0029","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[0,0],[1,1],[1,1]],\"zero_index\":0}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [1, 1], [1, 1]], 0, [0, 0], 是, [1, 0, 0], 是)","explanation_zh":"收束波核對=([[0, 0], [1, 1], [1, 1]], 0, [0, 0], 是, [1, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254029,"parameters":{"vectors":[[0,0],[1,1],[1,1]],"zero_index":0,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[0,0],[1,1],[1,1]],\"zero_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0030","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,-8],[0,0],[1,1]],\"zero_index\":1}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, -8], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)","explanation_zh":"收束波核對=([[1, -8], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254030,"parameters":{"vectors":[[1,-8],[0,0],[1,1]],"zero_index":1,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-8],[0,0],[1,1]],\"zero_index\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0031","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,-7],[-2,1],[0,0]],\"zero_index\":2}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, -7], [-2, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)","explanation_zh":"收束波核對=([[1, -7], [-2, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254031,"parameters":{"vectors":[[1,-7],[-2,1],[0,0]],"zero_index":2,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-7],[-2,1],[0,0]],\"zero_index\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0032","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,0],[-1,1],[1,1]],\"zero_index\":0}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [-1, 1], [1, 1]], 0, [1, 0], 否, [1, 0, 0], 否)","explanation_zh":"收束波核對=([[1, 0], [-1, 1], [1, 1]], 0, [1, 0], 否, [1, 0, 0], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254032,"parameters":{"vectors":[[1,0],[-1,1],[1,1]],"zero_index":0,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,0],[-1,1],[1,1]],\"zero_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0033","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,-5],[0,0],[1,1]],\"zero_index\":1}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, -5], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)","explanation_zh":"收束波核對=([[1, -5], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254033,"parameters":{"vectors":[[1,-5],[0,0],[1,1]],"zero_index":1,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-5],[0,0],[1,1]],\"zero_index\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0034","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,-4],[1,1],[0,0]],\"zero_index\":2}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, -4], [1, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)","explanation_zh":"收束波核對=([[1, -4], [1, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254034,"parameters":{"vectors":[[1,-4],[1,1],[0,0]],"zero_index":2,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-4],[1,1],[0,0]],\"zero_index\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0035","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[0,0],[2,1],[1,1]],\"zero_index\":0}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [2, 1], [1, 1]], 0, [0, 0], 是, [1, 0, 0], 是)","explanation_zh":"收束波核對=([[0, 0], [2, 1], [1, 1]], 0, [0, 0], 是, [1, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254035,"parameters":{"vectors":[[0,0],[2,1],[1,1]],"zero_index":0,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[0,0],[2,1],[1,1]],\"zero_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0036","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,-2],[0,0],[1,1]],\"zero_index\":1}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, -2], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)","explanation_zh":"收束波核對=([[1, -2], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254036,"parameters":{"vectors":[[1,-2],[0,0],[1,1]],"zero_index":1,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-2],[0,0],[1,1]],\"zero_index\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0037","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,-1],[-1,1],[0,0]],\"zero_index\":2}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, -1], [-1, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)","explanation_zh":"收束波核對=([[1, -1], [-1, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254037,"parameters":{"vectors":[[1,-1],[-1,1],[0,0]],"zero_index":2,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-1],[-1,1],[0,0]],\"zero_index\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0038","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"zero_index\":0}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], 0, [1, 0], 否, [1, 0, 0], 否)","explanation_zh":"收束波核對=([[1, 0], [0, 1], [1, 1]], 0, [1, 0], 否, [1, 0, 0], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254038,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"zero_index":0,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,0],[0,1],[1,1]],\"zero_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0039","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,1],[0,0],[1,1]],\"zero_index\":1}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)","explanation_zh":"收束波核對=([[1, 1], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254039,"parameters":{"vectors":[[1,1],[0,0],[1,1]],"zero_index":1,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,1],[0,0],[1,1]],\"zero_index\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0040","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,2],[2,1],[0,0]],\"zero_index\":2}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)","explanation_zh":"收束波核對=([[1, 2], [2, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254040,"parameters":{"vectors":[[1,2],[2,1],[0,0]],"zero_index":2,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,2],[2,1],[0,0]],\"zero_index\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0041","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[0,0],[-2,1],[1,1]],\"zero_index\":0}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [-2, 1], [1, 1]], 0, [0, 0], 是, [1, 0, 0], 是)","explanation_zh":"收束波核對=([[0, 0], [-2, 1], [1, 1]], 0, [0, 0], 是, [1, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254041,"parameters":{"vectors":[[0,0],[-2,1],[1,1]],"zero_index":0,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[0,0],[-2,1],[1,1]],\"zero_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0042","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,4],[0,0],[1,1]],\"zero_index\":1}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 4], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)","explanation_zh":"收束波核對=([[1, 4], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254042,"parameters":{"vectors":[[1,4],[0,0],[1,1]],"zero_index":1,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,4],[0,0],[1,1]],\"zero_index\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0043","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,5],[0,1],[0,0]],\"zero_index\":2}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 5], [0, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)","explanation_zh":"收束波核對=([[1, 5], [0, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254043,"parameters":{"vectors":[[1,5],[0,1],[0,0]],"zero_index":2,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,5],[0,1],[0,0]],\"zero_index\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0044","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,0],[1,1],[1,1]],\"zero_index\":0}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [1, 1], [1, 1]], 0, [1, 0], 否, [1, 0, 0], 否)","explanation_zh":"收束波核對=([[1, 0], [1, 1], [1, 1]], 0, [1, 0], 否, [1, 0, 0], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254044,"parameters":{"vectors":[[1,0],[1,1],[1,1]],"zero_index":0,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,0],[1,1],[1,1]],\"zero_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0045","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,7],[0,0],[1,1]],\"zero_index\":1}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 7], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)","explanation_zh":"收束波核對=([[1, 7], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254045,"parameters":{"vectors":[[1,7],[0,0],[1,1]],"zero_index":1,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,7],[0,0],[1,1]],\"zero_index\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0046","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,8],[-2,1],[0,0]],\"zero_index\":2}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 8], [-2, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)","explanation_zh":"收束波核對=([[1, 8], [-2, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254046,"parameters":{"vectors":[[1,8],[-2,1],[0,0]],"zero_index":2,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,8],[-2,1],[0,0]],\"zero_index\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0047","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[0,0],[-1,1],[1,1]],\"zero_index\":0}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [-1, 1], [1, 1]], 0, [0, 0], 是, [1, 0, 0], 是)","explanation_zh":"收束波核對=([[0, 0], [-1, 1], [1, 1]], 0, [0, 0], 是, [1, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254047,"parameters":{"vectors":[[0,0],[-1,1],[1,1]],"zero_index":0,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[0,0],[-1,1],[1,1]],\"zero_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0048","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,10],[0,0],[1,1]],\"zero_index\":1}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 10], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)","explanation_zh":"收束波核對=([[1, 10], [0, 0], [1, 1]], 1, [0, 0], 是, [0, 1, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254048,"parameters":{"vectors":[[1,10],[0,0],[1,1]],"zero_index":1,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,10],[0,0],[1,1]],\"zero_index\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0049","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,11],[1,1],[0,0]],\"zero_index\":2}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 11], [1, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)","explanation_zh":"收束波核對=([[1, 11], [1, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254049,"parameters":{"vectors":[[1,11],[1,1],[0,0]],"zero_index":2,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,11],[1,1],[0,0]],\"zero_index\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0050","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={\"vectors\":[[1,0],[2,1],[1,1]],\"zero_index\":0}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [2, 1], [1, 1]], 0, [1, 0], 否, [1, 0, 0], 否)","explanation_zh":"收束波核對=([[1, 0], [2, 1], [1, 1]], 0, [1, 0], 否, [1, 0, 0], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254050,"parameters":{"vectors":[[1,0],[2,1],[1,1]],"zero_index":0,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-zero-vector-dependence-certificate","semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,0],[2,1],[1,1]],\"zero_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0051","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[1,2],\"right\":[1,3],\"scalar\":1}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [1, 3], 1, [1, 2], [0, -1], 否)","explanation_zh":"收束波核對=([1, 2], [1, 3], 1, [1, 2], [0, -1], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254051,"parameters":{"left":[1,2],"right":[1,3],"scalar":1,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[1,2],\"right\":[1,3],\"scalar\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0052","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[2,3],\"right\":[4,6],\"scalar\":2}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([2, 3], [4, 6], 2, [4, 6], [0, 0], 是)","explanation_zh":"收束波核對=([2, 3], [4, 6], 2, [4, 6], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254052,"parameters":{"left":[2,3],"right":[4,6],"scalar":2,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[2,3],\"right\":[4,6],\"scalar\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0053","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[3,4],\"right\":[9,12],\"scalar\":3}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([3, 4], [9, 12], 3, [9, 12], [0, 0], 是)","explanation_zh":"收束波核對=([3, 4], [9, 12], 3, [9, 12], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254053,"parameters":{"left":[3,4],"right":[9,12],"scalar":3,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[3,4],\"right\":[9,12],\"scalar\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0054","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[4,5],\"right\":[16,20],\"scalar\":4}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([4, 5], [16, 20], 4, [16, 20], [0, 0], 是)","explanation_zh":"收束波核對=([4, 5], [16, 20], 4, [16, 20], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254054,"parameters":{"left":[4,5],"right":[16,20],"scalar":4,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[4,5],\"right\":[16,20],\"scalar\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0055","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[5,6],\"right\":[25,30],\"scalar\":5}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([5, 6], [25, 30], 5, [25, 30], [0, 0], 是)","explanation_zh":"收束波核對=([5, 6], [25, 30], 5, [25, 30], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254055,"parameters":{"left":[5,6],"right":[25,30],"scalar":5,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[5,6],\"right\":[25,30],\"scalar\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0056","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[6,7],\"right\":[6,7],\"scalar\":1}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([6, 7], [6, 7], 1, [6, 7], [0, 0], 是)","explanation_zh":"收束波核對=([6, 7], [6, 7], 1, [6, 7], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254056,"parameters":{"left":[6,7],"right":[6,7],"scalar":1,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[6,7],\"right\":[6,7],\"scalar\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0057","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[7,8],\"right\":[14,17],\"scalar\":2}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([7, 8], [14, 17], 2, [14, 16], [0, -1], 否)","explanation_zh":"收束波核對=([7, 8], [14, 17], 2, [14, 16], [0, -1], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254057,"parameters":{"left":[7,8],"right":[14,17],"scalar":2,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[7,8],\"right\":[14,17],\"scalar\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0058","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[8,9],\"right\":[24,27],\"scalar\":3}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([8, 9], [24, 27], 3, [24, 27], [0, 0], 是)","explanation_zh":"收束波核對=([8, 9], [24, 27], 3, [24, 27], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254058,"parameters":{"left":[8,9],"right":[24,27],"scalar":3,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[8,9],\"right\":[24,27],\"scalar\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0059","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[9,10],\"right\":[36,40],\"scalar\":4}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([9, 10], [36, 40], 4, [36, 40], [0, 0], 是)","explanation_zh":"收束波核對=([9, 10], [36, 40], 4, [36, 40], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254059,"parameters":{"left":[9,10],"right":[36,40],"scalar":4,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[9,10],\"right\":[36,40],\"scalar\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0060","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[10,11],\"right\":[50,55],\"scalar\":5}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([10, 11], [50, 55], 5, [50, 55], [0, 0], 是)","explanation_zh":"收束波核對=([10, 11], [50, 55], 5, [50, 55], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254060,"parameters":{"left":[10,11],"right":[50,55],"scalar":5,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[10,11],\"right\":[50,55],\"scalar\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0061","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[11,12],\"right\":[11,12],\"scalar\":1}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([11, 12], [11, 12], 1, [11, 12], [0, 0], 是)","explanation_zh":"收束波核對=([11, 12], [11, 12], 1, [11, 12], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254061,"parameters":{"left":[11,12],"right":[11,12],"scalar":1,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[11,12],\"right\":[11,12],\"scalar\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0062","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[12,13],\"right\":[24,26],\"scalar\":2}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([12, 13], [24, 26], 2, [24, 26], [0, 0], 是)","explanation_zh":"收束波核對=([12, 13], [24, 26], 2, [24, 26], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254062,"parameters":{"left":[12,13],"right":[24,26],"scalar":2,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[12,13],\"right\":[24,26],\"scalar\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0063","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[13,14],\"right\":[39,43],\"scalar\":3}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([13, 14], [39, 43], 3, [39, 42], [0, -1], 否)","explanation_zh":"收束波核對=([13, 14], [39, 43], 3, [39, 42], [0, -1], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254063,"parameters":{"left":[13,14],"right":[39,43],"scalar":3,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[13,14],\"right\":[39,43],\"scalar\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0064","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[14,15],\"right\":[56,60],\"scalar\":4}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([14, 15], [56, 60], 4, [56, 60], [0, 0], 是)","explanation_zh":"收束波核對=([14, 15], [56, 60], 4, [56, 60], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254064,"parameters":{"left":[14,15],"right":[56,60],"scalar":4,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[14,15],\"right\":[56,60],\"scalar\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0065","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[15,16],\"right\":[75,80],\"scalar\":5}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([15, 16], [75, 80], 5, [75, 80], [0, 0], 是)","explanation_zh":"收束波核對=([15, 16], [75, 80], 5, [75, 80], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254065,"parameters":{"left":[15,16],"right":[75,80],"scalar":5,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[15,16],\"right\":[75,80],\"scalar\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0066","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[16,17],\"right\":[16,17],\"scalar\":1}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([16, 17], [16, 17], 1, [16, 17], [0, 0], 是)","explanation_zh":"收束波核對=([16, 17], [16, 17], 1, [16, 17], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254066,"parameters":{"left":[16,17],"right":[16,17],"scalar":1,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[16,17],\"right\":[16,17],\"scalar\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0067","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[17,18],\"right\":[34,36],\"scalar\":2}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([17, 18], [34, 36], 2, [34, 36], [0, 0], 是)","explanation_zh":"收束波核對=([17, 18], [34, 36], 2, [34, 36], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254067,"parameters":{"left":[17,18],"right":[34,36],"scalar":2,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[17,18],\"right\":[34,36],\"scalar\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0068","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[18,19],\"right\":[54,57],\"scalar\":3}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([18, 19], [54, 57], 3, [54, 57], [0, 0], 是)","explanation_zh":"收束波核對=([18, 19], [54, 57], 3, [54, 57], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254068,"parameters":{"left":[18,19],"right":[54,57],"scalar":3,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[18,19],\"right\":[54,57],\"scalar\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0069","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[19,20],\"right\":[76,81],\"scalar\":4}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([19, 20], [76, 81], 4, [76, 80], [0, -1], 否)","explanation_zh":"收束波核對=([19, 20], [76, 81], 4, [76, 80], [0, -1], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254069,"parameters":{"left":[19,20],"right":[76,81],"scalar":4,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[19,20],\"right\":[76,81],\"scalar\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0070","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[20,21],\"right\":[100,105],\"scalar\":5}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([20, 21], [100, 105], 5, [100, 105], [0, 0], 是)","explanation_zh":"收束波核對=([20, 21], [100, 105], 5, [100, 105], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254070,"parameters":{"left":[20,21],"right":[100,105],"scalar":5,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[20,21],\"right\":[100,105],\"scalar\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0071","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[21,22],\"right\":[21,22],\"scalar\":1}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([21, 22], [21, 22], 1, [21, 22], [0, 0], 是)","explanation_zh":"收束波核對=([21, 22], [21, 22], 1, [21, 22], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254071,"parameters":{"left":[21,22],"right":[21,22],"scalar":1,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[21,22],\"right\":[21,22],\"scalar\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0072","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[22,23],\"right\":[44,46],\"scalar\":2}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([22, 23], [44, 46], 2, [44, 46], [0, 0], 是)","explanation_zh":"收束波核對=([22, 23], [44, 46], 2, [44, 46], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254072,"parameters":{"left":[22,23],"right":[44,46],"scalar":2,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[22,23],\"right\":[44,46],\"scalar\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0073","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[23,24],\"right\":[69,72],\"scalar\":3}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([23, 24], [69, 72], 3, [69, 72], [0, 0], 是)","explanation_zh":"收束波核對=([23, 24], [69, 72], 3, [69, 72], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254073,"parameters":{"left":[23,24],"right":[69,72],"scalar":3,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[23,24],\"right\":[69,72],\"scalar\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0074","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[24,25],\"right\":[96,100],\"scalar\":4}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([24, 25], [96, 100], 4, [96, 100], [0, 0], 是)","explanation_zh":"收束波核對=([24, 25], [96, 100], 4, [96, 100], [0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254074,"parameters":{"left":[24,25],"right":[96,100],"scalar":4,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[24,25],\"right\":[96,100],\"scalar\":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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0075","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-proportional-pair-dependence-certificate；資料={\"left\":[25,26],\"right\":[125,131],\"scalar\":5}。請依 [left,right,scalar,scaled_left,relation,dependence_certificate] 欄序回答。","answer":{"kind":"expression","value":"([25, 26], [125, 131], 5, [125, 130], [0, -1], 否)","explanation_zh":"收束波核對=([25, 26], [125, 131], 5, [125, 130], [0, -1], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254075,"parameters":{"left":[25,26],"right":[125,131],"scalar":5,"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-proportional-pair-dependence-certificate","semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[25,26],\"right\":[125,131],\"scalar\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0076","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,-2],[0,1]],\"target\":[-12,21],\"first_coordinates\":[-12,-3],\"second_coordinates\":[-11,-3]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [-12, 21], 1, [-12, -3], [-12, 21], 是, [-11, -3], [-11, 19], 否, 是)","explanation_zh":"收束波核對=([[1, -2], [0, 1]], [-12, 21], 1, [-12, -3], [-12, 21], 是, [-11, -3], [-11, 19], 否, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254076,"parameters":{"basis":[[1,-2],[0,1]],"target":[-12,21],"first_coordinates":[-12,-3],"second_coordinates":[-11,-3],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-2],[0,1]],\"first_coordinates\":[-12,-3],\"second_coordinates\":[-11,-3],\"target\":[-12,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0077","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,-1],[0,1]],\"target\":[-11,9],\"first_coordinates\":[-11,-2],\"second_coordinates\":[-11,-2]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [-11, 9], 1, [-11, -2], [-11, 9], 是, [-11, -2], [-11, 9], 是, 是)","explanation_zh":"收束波核對=([[1, -1], [0, 1]], [-11, 9], 1, [-11, -2], [-11, 9], 是, [-11, -2], [-11, 9], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254077,"parameters":{"basis":[[1,-1],[0,1]],"target":[-11,9],"first_coordinates":[-11,-2],"second_coordinates":[-11,-2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-1],[0,1]],\"first_coordinates\":[-11,-2],\"second_coordinates\":[-11,-2],\"target\":[-11,9]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0078","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,0],[0,1]],\"target\":[-10,-1],\"first_coordinates\":[-10,-1],\"second_coordinates\":[-10,-1]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-10, -1], 1, [-10, -1], [-10, -1], 是, [-10, -1], [-10, -1], 是, 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [-10, -1], 1, [-10, -1], [-10, -1], 是, [-10, -1], [-10, -1], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254078,"parameters":{"basis":[[1,0],[0,1]],"target":[-10,-1],"first_coordinates":[-10,-1],"second_coordinates":[-10,-1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,0],[0,1]],\"first_coordinates\":[-10,-1],\"second_coordinates\":[-10,-1],\"target\":[-10,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0079","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,1],[0,1]],\"target\":[-9,-9],\"first_coordinates\":[-9,0],\"second_coordinates\":[-9,0]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [-9, -9], 1, [-9, 0], [-9, -9], 是, [-9, 0], [-9, -9], 是, 是)","explanation_zh":"收束波核對=([[1, 1], [0, 1]], [-9, -9], 1, [-9, 0], [-9, -9], 是, [-9, 0], [-9, -9], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254079,"parameters":{"basis":[[1,1],[0,1]],"target":[-9,-9],"first_coordinates":[-9,0],"second_coordinates":[-9,0],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,1],[0,1]],\"first_coordinates\":[-9,0],\"second_coordinates\":[-9,0],\"target\":[-9,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0080","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,2],[0,1]],\"target\":[-8,-15],\"first_coordinates\":[-8,1],\"second_coordinates\":[-7,1]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [-8, -15], 1, [-8, 1], [-8, -15], 是, [-7, 1], [-7, -13], 否, 是)","explanation_zh":"收束波核對=([[1, 2], [0, 1]], [-8, -15], 1, [-8, 1], [-8, -15], 是, [-7, 1], [-7, -13], 否, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254080,"parameters":{"basis":[[1,2],[0,1]],"target":[-8,-15],"first_coordinates":[-8,1],"second_coordinates":[-7,1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,2],[0,1]],\"first_coordinates\":[-8,1],\"second_coordinates\":[-7,1],\"target\":[-8,-15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0081","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,-2],[0,1]],\"target\":[-7,16],\"first_coordinates\":[-7,2],\"second_coordinates\":[-7,2]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [-7, 16], 1, [-7, 2], [-7, 16], 是, [-7, 2], [-7, 16], 是, 是)","explanation_zh":"收束波核對=([[1, -2], [0, 1]], [-7, 16], 1, [-7, 2], [-7, 16], 是, [-7, 2], [-7, 16], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254081,"parameters":{"basis":[[1,-2],[0,1]],"target":[-7,16],"first_coordinates":[-7,2],"second_coordinates":[-7,2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-2],[0,1]],\"first_coordinates\":[-7,2],\"second_coordinates\":[-7,2],\"target\":[-7,16]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0082","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,-1],[0,1]],\"target\":[-6,9],\"first_coordinates\":[-6,3],\"second_coordinates\":[-6,3]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [-6, 9], 1, [-6, 3], [-6, 9], 是, [-6, 3], [-6, 9], 是, 是)","explanation_zh":"收束波核對=([[1, -1], [0, 1]], [-6, 9], 1, [-6, 3], [-6, 9], 是, [-6, 3], [-6, 9], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254082,"parameters":{"basis":[[1,-1],[0,1]],"target":[-6,9],"first_coordinates":[-6,3],"second_coordinates":[-6,3],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-1],[0,1]],\"first_coordinates\":[-6,3],\"second_coordinates\":[-6,3],\"target\":[-6,9]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0083","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,0],[0,1]],\"target\":[-5,-3],\"first_coordinates\":[-5,-3],\"second_coordinates\":[-5,-3]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-5, -3], 1, [-5, -3], [-5, -3], 是, [-5, -3], [-5, -3], 是, 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [-5, -3], 1, [-5, -3], [-5, -3], 是, [-5, -3], [-5, -3], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254083,"parameters":{"basis":[[1,0],[0,1]],"target":[-5,-3],"first_coordinates":[-5,-3],"second_coordinates":[-5,-3],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,0],[0,1]],\"first_coordinates\":[-5,-3],\"second_coordinates\":[-5,-3],\"target\":[-5,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0084","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,1],[0,1]],\"target\":[-4,-6],\"first_coordinates\":[-4,-2],\"second_coordinates\":[-3,-2]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [-4, -6], 1, [-4, -2], [-4, -6], 是, [-3, -2], [-3, -5], 否, 是)","explanation_zh":"收束波核對=([[1, 1], [0, 1]], [-4, -6], 1, [-4, -2], [-4, -6], 是, [-3, -2], [-3, -5], 否, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254084,"parameters":{"basis":[[1,1],[0,1]],"target":[-4,-6],"first_coordinates":[-4,-2],"second_coordinates":[-3,-2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,1],[0,1]],\"first_coordinates\":[-4,-2],\"second_coordinates\":[-3,-2],\"target\":[-4,-6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0085","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,2],[0,1]],\"target\":[-3,-7],\"first_coordinates\":[-3,-1],\"second_coordinates\":[-3,-1]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [-3, -7], 1, [-3, -1], [-3, -7], 是, [-3, -1], [-3, -7], 是, 是)","explanation_zh":"收束波核對=([[1, 2], [0, 1]], [-3, -7], 1, [-3, -1], [-3, -7], 是, [-3, -1], [-3, -7], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254085,"parameters":{"basis":[[1,2],[0,1]],"target":[-3,-7],"first_coordinates":[-3,-1],"second_coordinates":[-3,-1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,2],[0,1]],\"first_coordinates\":[-3,-1],\"second_coordinates\":[-3,-1],\"target\":[-3,-7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0086","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,-2],[0,1]],\"target\":[-2,4],\"first_coordinates\":[-2,0],\"second_coordinates\":[-2,0]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [-2, 4], 1, [-2, 0], [-2, 4], 是, [-2, 0], [-2, 4], 是, 是)","explanation_zh":"收束波核對=([[1, -2], [0, 1]], [-2, 4], 1, [-2, 0], [-2, 4], 是, [-2, 0], [-2, 4], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254086,"parameters":{"basis":[[1,-2],[0,1]],"target":[-2,4],"first_coordinates":[-2,0],"second_coordinates":[-2,0],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-2],[0,1]],\"first_coordinates\":[-2,0],\"second_coordinates\":[-2,0],\"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0087","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,-1],[0,1]],\"target\":[-1,2],\"first_coordinates\":[-1,1],\"second_coordinates\":[-1,1]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [-1, 2], 1, [-1, 1], [-1, 2], 是, [-1, 1], [-1, 2], 是, 是)","explanation_zh":"收束波核對=([[1, -1], [0, 1]], [-1, 2], 1, [-1, 1], [-1, 2], 是, [-1, 1], [-1, 2], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254087,"parameters":{"basis":[[1,-1],[0,1]],"target":[-1,2],"first_coordinates":[-1,1],"second_coordinates":[-1,1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-1],[0,1]],\"first_coordinates\":[-1,1],\"second_coordinates\":[-1,1],\"target\":[-1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0088","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,0],[0,1]],\"target\":[0,2],\"first_coordinates\":[0,2],\"second_coordinates\":[1,2]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [0, 2], 1, [0, 2], [0, 2], 是, [1, 2], [1, 2], 否, 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [0, 2], 1, [0, 2], [0, 2], 是, [1, 2], [1, 2], 否, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254088,"parameters":{"basis":[[1,0],[0,1]],"target":[0,2],"first_coordinates":[0,2],"second_coordinates":[1,2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,0],[0,1]],\"first_coordinates\":[0,2],\"second_coordinates\":[1,2],\"target\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0089","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,1],[0,1]],\"target\":[1,4],\"first_coordinates\":[1,3],\"second_coordinates\":[1,3]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [1, 4], 1, [1, 3], [1, 4], 是, [1, 3], [1, 4], 是, 是)","explanation_zh":"收束波核對=([[1, 1], [0, 1]], [1, 4], 1, [1, 3], [1, 4], 是, [1, 3], [1, 4], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254089,"parameters":{"basis":[[1,1],[0,1]],"target":[1,4],"first_coordinates":[1,3],"second_coordinates":[1,3],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,1],[0,1]],\"first_coordinates\":[1,3],\"second_coordinates\":[1,3],\"target\":[1,4]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0090","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,2],[0,1]],\"target\":[2,1],\"first_coordinates\":[2,-3],\"second_coordinates\":[2,-3]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [2, 1], 1, [2, -3], [2, 1], 是, [2, -3], [2, 1], 是, 是)","explanation_zh":"收束波核對=([[1, 2], [0, 1]], [2, 1], 1, [2, -3], [2, 1], 是, [2, -3], [2, 1], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254090,"parameters":{"basis":[[1,2],[0,1]],"target":[2,1],"first_coordinates":[2,-3],"second_coordinates":[2,-3],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,2],[0,1]],\"first_coordinates\":[2,-3],\"second_coordinates\":[2,-3],\"target\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0091","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,-2],[0,1]],\"target\":[3,-8],\"first_coordinates\":[3,-2],\"second_coordinates\":[3,-2]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [3, -8], 1, [3, -2], [3, -8], 是, [3, -2], [3, -8], 是, 是)","explanation_zh":"收束波核對=([[1, -2], [0, 1]], [3, -8], 1, [3, -2], [3, -8], 是, [3, -2], [3, -8], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254091,"parameters":{"basis":[[1,-2],[0,1]],"target":[3,-8],"first_coordinates":[3,-2],"second_coordinates":[3,-2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-2],[0,1]],\"first_coordinates\":[3,-2],\"second_coordinates\":[3,-2],\"target\":[3,-8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0092","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,-1],[0,1]],\"target\":[4,-5],\"first_coordinates\":[4,-1],\"second_coordinates\":[5,-1]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [4, -5], 1, [4, -1], [4, -5], 是, [5, -1], [5, -6], 否, 是)","explanation_zh":"收束波核對=([[1, -1], [0, 1]], [4, -5], 1, [4, -1], [4, -5], 是, [5, -1], [5, -6], 否, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254092,"parameters":{"basis":[[1,-1],[0,1]],"target":[4,-5],"first_coordinates":[4,-1],"second_coordinates":[5,-1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-1],[0,1]],\"first_coordinates\":[4,-1],\"second_coordinates\":[5,-1],\"target\":[4,-5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0093","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,0],[0,1]],\"target\":[5,0],\"first_coordinates\":[5,0],\"second_coordinates\":[5,0]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [5, 0], 1, [5, 0], [5, 0], 是, [5, 0], [5, 0], 是, 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [5, 0], 1, [5, 0], [5, 0], 是, [5, 0], [5, 0], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254093,"parameters":{"basis":[[1,0],[0,1]],"target":[5,0],"first_coordinates":[5,0],"second_coordinates":[5,0],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,0],[0,1]],\"first_coordinates\":[5,0],\"second_coordinates\":[5,0],\"target\":[5,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0094","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,1],[0,1]],\"target\":[6,7],\"first_coordinates\":[6,1],\"second_coordinates\":[6,1]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [6, 7], 1, [6, 1], [6, 7], 是, [6, 1], [6, 7], 是, 是)","explanation_zh":"收束波核對=([[1, 1], [0, 1]], [6, 7], 1, [6, 1], [6, 7], 是, [6, 1], [6, 7], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254094,"parameters":{"basis":[[1,1],[0,1]],"target":[6,7],"first_coordinates":[6,1],"second_coordinates":[6,1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,1],[0,1]],\"first_coordinates\":[6,1],\"second_coordinates\":[6,1],\"target\":[6,7]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0095","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,2],[0,1]],\"target\":[7,16],\"first_coordinates\":[7,2],\"second_coordinates\":[7,2]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [7, 16], 1, [7, 2], [7, 16], 是, [7, 2], [7, 16], 是, 是)","explanation_zh":"收束波核對=([[1, 2], [0, 1]], [7, 16], 1, [7, 2], [7, 16], 是, [7, 2], [7, 16], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254095,"parameters":{"basis":[[1,2],[0,1]],"target":[7,16],"first_coordinates":[7,2],"second_coordinates":[7,2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,2],[0,1]],\"first_coordinates\":[7,2],\"second_coordinates\":[7,2],\"target\":[7,16]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0096","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,-2],[0,1]],\"target\":[8,-13],\"first_coordinates\":[8,3],\"second_coordinates\":[9,3]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [8, -13], 1, [8, 3], [8, -13], 是, [9, 3], [9, -15], 否, 是)","explanation_zh":"收束波核對=([[1, -2], [0, 1]], [8, -13], 1, [8, 3], [8, -13], 是, [9, 3], [9, -15], 否, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254096,"parameters":{"basis":[[1,-2],[0,1]],"target":[8,-13],"first_coordinates":[8,3],"second_coordinates":[9,3],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-2],[0,1]],\"first_coordinates\":[8,3],\"second_coordinates\":[9,3],\"target\":[8,-13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0097","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,-1],[0,1]],\"target\":[9,-12],\"first_coordinates\":[9,-3],\"second_coordinates\":[9,-3]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [9, -12], 1, [9, -3], [9, -12], 是, [9, -3], [9, -12], 是, 是)","explanation_zh":"收束波核對=([[1, -1], [0, 1]], [9, -12], 1, [9, -3], [9, -12], 是, [9, -3], [9, -12], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254097,"parameters":{"basis":[[1,-1],[0,1]],"target":[9,-12],"first_coordinates":[9,-3],"second_coordinates":[9,-3],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-1],[0,1]],\"first_coordinates\":[9,-3],\"second_coordinates\":[9,-3],\"target\":[9,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0098","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,0],[0,1]],\"target\":[10,-2],\"first_coordinates\":[10,-2],\"second_coordinates\":[10,-2]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [10, -2], 1, [10, -2], [10, -2], 是, [10, -2], [10, -2], 是, 是)","explanation_zh":"收束波核對=([[1, 0], [0, 1]], [10, -2], 1, [10, -2], [10, -2], 是, [10, -2], [10, -2], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254098,"parameters":{"basis":[[1,0],[0,1]],"target":[10,-2],"first_coordinates":[10,-2],"second_coordinates":[10,-2],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,0],[0,1]],\"first_coordinates\":[10,-2],\"second_coordinates\":[10,-2],\"target\":[10,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0099","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,1],[0,1]],\"target\":[11,10],\"first_coordinates\":[11,-1],\"second_coordinates\":[11,-1]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [11, 10], 1, [11, -1], [11, 10], 是, [11, -1], [11, 10], 是, 是)","explanation_zh":"收束波核對=([[1, 1], [0, 1]], [11, 10], 1, [11, -1], [11, 10], 是, [11, -1], [11, 10], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254099,"parameters":{"basis":[[1,1],[0,1]],"target":[11,10],"first_coordinates":[11,-1],"second_coordinates":[11,-1],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,1],[0,1]],\"first_coordinates\":[11,-1],\"second_coordinates\":[11,-1],\"target\":[11,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r04-linear-independence-basis-0100","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"基底密度","stem_zh":"只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,2],[0,1]],\"target\":[12,24],\"first_coordinates\":[12,0],\"second_coordinates\":[13,0]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [12, 24], 1, [12, 0], [12, 24], 是, [13, 0], [13, 26], 否, 是)","explanation_zh":"收束波核對=([[1, 2], [0, 1]], [12, 24], 1, [12, 0], [12, 24], 是, [13, 0], [13, 26], 否, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底密度"]},"variant":{"family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254100,"parameters":{"basis":[[1,2],[0,1]],"target":[12,24],"first_coordinates":[12,0],"second_coordinates":[13,0],"domain":"R2_basis_density_without_dimension_extension","operation":"basisdeep-r2-coordinate-uniqueness-audit","semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,2],[0,1]],\"first_coordinates\":[12,0],\"second_coordinates\":[13,0],\"target\":[12,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、BVP適定性、全域最佳解、Markov長期性、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"}}