{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0001","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[0,0],[0,0]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, 0, 0)","explanation_zh":"降冪特徵多項式係數=(1, 0, 0)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097001,"parameters":{"M":[[0,0],[0,0]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"0\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0002","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[1,0],[0,1]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -2, 1)","explanation_zh":"降冪特徵多項式係數=(1, -2, 1)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097002,"parameters":{"M":[[1,0],[0,1]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"2\",\"1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0003","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[-10,3],[1,3]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, 7, -33)","explanation_zh":"降冪特徵多項式係數=(1, 7, -33)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097003,"parameters":{"M":[[-10,3],[1,3]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"-7\",\"-33\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0004","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[-9,1],[-1,4]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, 5, -35)","explanation_zh":"降冪特徵多項式係數=(1, 5, -35)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097004,"parameters":{"M":[[-9,1],[-1,4]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"-5\",\"-35\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0005","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[-8,2],[1,5]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, 3, -42)","explanation_zh":"降冪特徵多項式係數=(1, 3, -42)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097005,"parameters":{"M":[[-8,2],[1,5]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"-3\",\"-42\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0006","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[-7,3],[-1,6]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, 1, -39)","explanation_zh":"降冪特徵多項式係數=(1, 1, -39)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097006,"parameters":{"M":[[-7,3],[-1,6]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"-1\",\"-39\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0007","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[-6,1],[1,7]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -1, -43)","explanation_zh":"降冪特徵多項式係數=(1, -1, -43)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097007,"parameters":{"M":[[-6,1],[1,7]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"1\",\"-43\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0008","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[-5,2],[-1,8]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -3, -38)","explanation_zh":"降冪特徵多項式係數=(1, -3, -38)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097008,"parameters":{"M":[[-5,2],[-1,8]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"3\",\"-38\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0009","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[-4,3],[1,9]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -5, -39)","explanation_zh":"降冪特徵多項式係數=(1, -5, -39)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097009,"parameters":{"M":[[-4,3],[1,9]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"5\",\"-39\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0010","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[-3,1],[-1,10]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -7, -29)","explanation_zh":"降冪特徵多項式係數=(1, -7, -29)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097010,"parameters":{"M":[[-3,1],[-1,10]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"7\",\"-29\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0011","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[-2,2],[1,11]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -9, -24)","explanation_zh":"降冪特徵多項式係數=(1, -9, -24)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097011,"parameters":{"M":[[-2,2],[1,11]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"9\",\"-24\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0012","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[-1,3],[-1,12]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -11, -9)","explanation_zh":"降冪特徵多項式係數=(1, -11, -9)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097012,"parameters":{"M":[[-1,3],[-1,12]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"11\",\"-9\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0013","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[0,1],[1,13]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -13, -1)","explanation_zh":"降冪特徵多項式係數=(1, -13, -1)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097013,"parameters":{"M":[[0,1],[1,13]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"13\",\"-1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0014","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[1,2],[-1,14]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -15, 16)","explanation_zh":"降冪特徵多項式係數=(1, -15, 16)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097014,"parameters":{"M":[[1,2],[-1,14]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"15\",\"16\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0015","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[2,3],[1,15]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -17, 27)","explanation_zh":"降冪特徵多項式係數=(1, -17, 27)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097015,"parameters":{"M":[[2,3],[1,15]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"17\",\"27\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0016","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[3,1],[-1,16]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -19, 49)","explanation_zh":"降冪特徵多項式係數=(1, -19, 49)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097016,"parameters":{"M":[[3,1],[-1,16]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"19\",\"49\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0017","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[4,2],[1,17]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -21, 66)","explanation_zh":"降冪特徵多項式係數=(1, -21, 66)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097017,"parameters":{"M":[[4,2],[1,17]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"21\",\"66\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0018","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[5,3],[-1,18]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -23, 93)","explanation_zh":"降冪特徵多項式係數=(1, -23, 93)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097018,"parameters":{"M":[[5,3],[-1,18]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"23\",\"93\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0019","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[6,1],[1,19]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -25, 113)","explanation_zh":"降冪特徵多項式係數=(1, -25, 113)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097019,"parameters":{"M":[[6,1],[1,19]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"25\",\"113\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0020","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[7,2],[-1,20]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -27, 142)","explanation_zh":"降冪特徵多項式係數=(1, -27, 142)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097020,"parameters":{"M":[[7,2],[-1,20]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"27\",\"142\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0021","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[8,3],[1,21]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -29, 165)","explanation_zh":"降冪特徵多項式係數=(1, -29, 165)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097021,"parameters":{"M":[[8,3],[1,21]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"29\",\"165\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0022","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[9,1],[-1,22]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -31, 199)","explanation_zh":"降冪特徵多項式係數=(1, -31, 199)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097022,"parameters":{"M":[[9,1],[-1,22]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"31\",\"199\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0023","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[10,2],[1,23]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -33, 228)","explanation_zh":"降冪特徵多項式係數=(1, -33, 228)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097023,"parameters":{"M":[[10,2],[1,23]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"33\",\"228\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0024","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[11,3],[-1,24]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -35, 267)","explanation_zh":"降冪特徵多項式係數=(1, -35, 267)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097024,"parameters":{"M":[[11,3],[-1,24]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"35\",\"267\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0025","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階矩陣的降冪特徵多項式","stem_zh":"實二階方陣與精確整數資料。M=[[12,1],[1,25]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。","answer":{"kind":"expression","value":"(1, -37, 299)","explanation_zh":"降冪特徵多項式係數=(1, -37, 299)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階矩陣的降冪特徵多項式"]},"variant":{"family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097025,"parameters":{"M":[[12,1],[1,25]],"operation":"density_eigen_characteristic_coefficients","domain":"real_two_by_two","semantic_template":"qf-density-eigen-characteristic-coefficients/[\"37\",\"299\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0026","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[0,-1],[1,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"-4","explanation_zh":"精確判別式=-4。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097026,"parameters":{"M":[[0,-1],[1,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"0\",\"-1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0027","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[3,1],[0,3]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"0","explanation_zh":"精確判別式=0。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097027,"parameters":{"M":[[3,1],[0,3]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"0\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0028","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[2,1],[2,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"12","explanation_zh":"精確判別式=12。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097028,"parameters":{"M":[[2,1],[2,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"2\",\"2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0029","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[3,1],[-3,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"-3","explanation_zh":"精確判別式=-3。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097029,"parameters":{"M":[[3,1],[-3,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"3\",\"-3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0030","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[4,1],[4,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"32","explanation_zh":"精確判別式=32。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097030,"parameters":{"M":[[4,1],[4,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"4\",\"4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0031","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[5,1],[-5,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"5","explanation_zh":"精確判別式=5。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097031,"parameters":{"M":[[5,1],[-5,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"5\",\"-5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0032","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[6,1],[6,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"60","explanation_zh":"精確判別式=60。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097032,"parameters":{"M":[[6,1],[6,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"6\",\"6\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0033","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[7,1],[-7,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"21","explanation_zh":"精確判別式=21。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097033,"parameters":{"M":[[7,1],[-7,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"7\",\"-7\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0034","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[8,1],[8,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"96","explanation_zh":"精確判別式=96。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097034,"parameters":{"M":[[8,1],[8,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"8\",\"8\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0035","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[9,1],[-9,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"45","explanation_zh":"精確判別式=45。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097035,"parameters":{"M":[[9,1],[-9,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"9\",\"-9\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0036","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[10,1],[10,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"140","explanation_zh":"精確判別式=140。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097036,"parameters":{"M":[[10,1],[10,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"10\",\"10\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0037","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[11,1],[-11,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"77","explanation_zh":"精確判別式=77。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097037,"parameters":{"M":[[11,1],[-11,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"11\",\"-11\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0038","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[12,1],[12,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"192","explanation_zh":"精確判別式=192。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097038,"parameters":{"M":[[12,1],[12,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"12\",\"12\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0039","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[13,1],[-13,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"117","explanation_zh":"精確判別式=117。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097039,"parameters":{"M":[[13,1],[-13,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"13\",\"-13\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0040","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[14,1],[14,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"252","explanation_zh":"精確判別式=252。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097040,"parameters":{"M":[[14,1],[14,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"14\",\"14\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0041","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[15,1],[-15,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"165","explanation_zh":"精確判別式=165。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097041,"parameters":{"M":[[15,1],[-15,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"15\",\"-15\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0042","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[16,1],[16,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"320","explanation_zh":"精確判別式=320。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097042,"parameters":{"M":[[16,1],[16,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"16\",\"16\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0043","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[17,1],[-17,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"221","explanation_zh":"精確判別式=221。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097043,"parameters":{"M":[[17,1],[-17,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"17\",\"-17\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0044","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[18,1],[18,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"396","explanation_zh":"精確判別式=396。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097044,"parameters":{"M":[[18,1],[18,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"18\",\"18\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0045","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[19,1],[-19,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"285","explanation_zh":"精確判別式=285。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097045,"parameters":{"M":[[19,1],[-19,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"19\",\"-19\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0046","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[20,1],[20,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"480","explanation_zh":"精確判別式=480。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097046,"parameters":{"M":[[20,1],[20,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"20\",\"20\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0047","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[21,1],[-21,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"357","explanation_zh":"精確判別式=357。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097047,"parameters":{"M":[[21,1],[-21,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"21\",\"-21\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0048","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[22,1],[22,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"572","explanation_zh":"精確判別式=572。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097048,"parameters":{"M":[[22,1],[22,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"22\",\"22\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0049","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[23,1],[-23,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"437","explanation_zh":"精確判別式=437。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097049,"parameters":{"M":[[23,1],[-23,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"23\",\"-23\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0050","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"實二階特徵多項式的精確判別式","stem_zh":"實二階方陣與精確整數資料。M=[[24,1],[24,0]]。求特徵多項式判別式Δ的整數值，不取浮點平方根。","answer":{"kind":"short_text","value":"672","explanation_zh":"精確判別式=672。產生端用(a−d)²+4bc另算；可以負、零或正，不補虛構實根。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實二階特徵多項式的精確判別式"]},"variant":{"family_id":"qf-density-eigen-discriminant","seed":202610097050,"parameters":{"M":[[24,1],[24,0]],"operation":"density_eigen_discriminant","domain":"real_two_by_two","semantic_template":"qf-density-eigen-discriminant/[\"24\",\"24\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0051","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[2,1],[0,-3]]，λ=4，v=[0, 0]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"核殘差=(0, 0)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097051,"parameters":{"M":[[2,1],[0,-3]],"lambda":4,"v":[0,0],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[\"zero_vector_residual\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0052","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[2,0],[0,3]]，λ=2，v=[1, 0]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"核殘差=(0, 0)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097052,"parameters":{"M":[[2,0],[0,3]],"lambda":2,"v":[1,0],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[2,0],[0,3]],2,[1,0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0053","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[3,3],[1,0]]，λ=0，v=[3, 0]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(9, 3)","explanation_zh":"核殘差=(9, 3)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097053,"parameters":{"M":[[3,3],[1,0]],"lambda":0,"v":[3,0],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[3,3],[1,0]],0,[3,0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0054","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[4,1],[-1,-1]]，λ=1，v=[1, -1]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(2, 1)","explanation_zh":"核殘差=(2, 1)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097054,"parameters":{"M":[[4,1],[-1,-1]],"lambda":1,"v":[1,-1],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[4,1],[-1,-1]],1,[1,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0055","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[5,2],[1,-2]]，λ=2，v=[2, -2]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(2, 10)","explanation_zh":"核殘差=(2, 10)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097055,"parameters":{"M":[[5,2],[1,-2]],"lambda":2,"v":[2,-2],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[5,2],[1,-2]],2,[2,-2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0056","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[6,3],[-1,-3]]，λ=-2，v=[3, 2]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(30, -5)","explanation_zh":"核殘差=(30, -5)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097056,"parameters":{"M":[[6,3],[-1,-3]],"lambda":-2,"v":[3,2],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[6,3],[-1,-3]],-2,[3,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0057","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[7,1],[1,-4]]，λ=-1，v=[1, 1]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(9, -2)","explanation_zh":"核殘差=(9, -2)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097057,"parameters":{"M":[[7,1],[1,-4]],"lambda":-1,"v":[1,1],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[7,1],[1,-4]],-1,[1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0058","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[8,2],[-1,-5]]，λ=0，v=[2, 0]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(16, -2)","explanation_zh":"核殘差=(16, -2)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097058,"parameters":{"M":[[8,2],[-1,-5]],"lambda":0,"v":[2,0],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[8,2],[-1,-5]],0,[2,0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0059","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[9,3],[1,-6]]，λ=1，v=[3, -1]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(21, 10)","explanation_zh":"核殘差=(21, 10)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097059,"parameters":{"M":[[9,3],[1,-6]],"lambda":1,"v":[3,-1],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[9,3],[1,-6]],1,[3,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0060","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[10,1],[-1,-7]]，λ=2，v=[1, -2]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(6, 17)","explanation_zh":"核殘差=(6, 17)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097060,"parameters":{"M":[[10,1],[-1,-7]],"lambda":2,"v":[1,-2],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[10,1],[-1,-7]],2,[1,-2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0061","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[11,2],[1,-8]]，λ=-2，v=[2, 2]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(30, -10)","explanation_zh":"核殘差=(30, -10)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097061,"parameters":{"M":[[11,2],[1,-8]],"lambda":-2,"v":[2,2],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[11,2],[1,-8]],-2,[2,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0062","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[12,3],[-1,-9]]，λ=-1，v=[3, 1]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(42, -11)","explanation_zh":"核殘差=(42, -11)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097062,"parameters":{"M":[[12,3],[-1,-9]],"lambda":-1,"v":[3,1],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[12,3],[-1,-9]],-1,[3,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0063","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[13,1],[1,-10]]，λ=0，v=[1, 0]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(13, 1)","explanation_zh":"核殘差=(13, 1)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097063,"parameters":{"M":[[13,1],[1,-10]],"lambda":0,"v":[1,0],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[13,1],[1,-10]],0,[1,0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0064","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[14,2],[-1,-11]]，λ=1，v=[2, -1]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(24, 10)","explanation_zh":"核殘差=(24, 10)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097064,"parameters":{"M":[[14,2],[-1,-11]],"lambda":1,"v":[2,-1],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[14,2],[-1,-11]],1,[2,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0065","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[15,3],[1,-12]]，λ=2，v=[3, -2]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(33, 31)","explanation_zh":"核殘差=(33, 31)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097065,"parameters":{"M":[[15,3],[1,-12]],"lambda":2,"v":[3,-2],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[15,3],[1,-12]],2,[3,-2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0066","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[16,1],[-1,-13]]，λ=-2，v=[1, 2]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(20, -23)","explanation_zh":"核殘差=(20, -23)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097066,"parameters":{"M":[[16,1],[-1,-13]],"lambda":-2,"v":[1,2],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[16,1],[-1,-13]],-2,[1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0067","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[17,2],[1,-14]]，λ=-1，v=[2, 1]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(38, -11)","explanation_zh":"核殘差=(38, -11)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097067,"parameters":{"M":[[17,2],[1,-14]],"lambda":-1,"v":[2,1],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[17,2],[1,-14]],-1,[2,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0068","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[18,3],[-1,-15]]，λ=0，v=[3, 0]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(54, -3)","explanation_zh":"核殘差=(54, -3)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097068,"parameters":{"M":[[18,3],[-1,-15]],"lambda":0,"v":[3,0],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[18,3],[-1,-15]],0,[3,0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0069","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[19,1],[1,-16]]，λ=1，v=[1, -1]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(17, 18)","explanation_zh":"核殘差=(17, 18)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097069,"parameters":{"M":[[19,1],[1,-16]],"lambda":1,"v":[1,-1],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[19,1],[1,-16]],1,[1,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0070","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[20,2],[-1,-17]]，λ=2，v=[2, -2]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(32, 36)","explanation_zh":"核殘差=(32, 36)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097070,"parameters":{"M":[[20,2],[-1,-17]],"lambda":2,"v":[2,-2],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[20,2],[-1,-17]],2,[2,-2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0071","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[21,3],[1,-18]]，λ=-2，v=[3, 2]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(75, -29)","explanation_zh":"核殘差=(75, -29)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097071,"parameters":{"M":[[21,3],[1,-18]],"lambda":-2,"v":[3,2],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[21,3],[1,-18]],-2,[3,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0072","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[22,1],[-1,-19]]，λ=-1，v=[1, 1]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(24, -19)","explanation_zh":"核殘差=(24, -19)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097072,"parameters":{"M":[[22,1],[-1,-19]],"lambda":-1,"v":[1,1],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[22,1],[-1,-19]],-1,[1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0073","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[23,2],[1,-20]]，λ=0，v=[2, 0]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(46, 2)","explanation_zh":"核殘差=(46, 2)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097073,"parameters":{"M":[[23,2],[1,-20]],"lambda":0,"v":[2,0],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[23,2],[1,-20]],0,[2,0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0074","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[24,3],[-1,-21]]，λ=1，v=[3, -1]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(66, 19)","explanation_zh":"核殘差=(66, 19)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097074,"parameters":{"M":[[24,3],[-1,-21]],"lambda":1,"v":[3,-1],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[24,3],[-1,-21]],1,[3,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0075","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"指定核方程的有序殘差","stem_zh":"實二階方陣與精確整數資料。M=[[25,1],[1,-22]]，λ=2，v=[1, -2]。列Mv−λv的兩個整數座標，保留順序。","answer":{"kind":"expression","value":"(21, 49)","explanation_zh":"核殘差=(21, 49)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定核方程的有序殘差"]},"variant":{"family_id":"qf-density-eigen-kernel-residual","seed":202610097075,"parameters":{"M":[[25,1],[1,-22]],"lambda":2,"v":[1,-2],"operation":"density_eigen_kernel_residual","domain":"real_two_by_two","semantic_template":"qf-density-eigen-kernel-residual/[[[25,1],[1,-22]],2,[1,-2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0076","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[1,1],[0,2]]，λ=2，非零v=[1, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"無","explanation_zh":"非零向量殘差=(0, 0)；第一失配=無。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097076,"parameters":{"M":[[1,1],[0,2]],"lambda":2,"v":[1,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,2]],2,[1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0077","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[0,2],[1,2]]，λ=0，非零v=[0, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"1","explanation_zh":"非零向量殘差=(2, 2)；第一失配=1。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097077,"parameters":{"M":[[0,2],[1,2]],"lambda":0,"v":[0,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[0,2],[1,2]],0,[0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0078","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[3,0],[0,8]]，λ=3，非零v=[1, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"2","explanation_zh":"非零向量殘差=(0, 5)；第一失配=2。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097078,"parameters":{"M":[[3,0],[0,8]],"lambda":3,"v":[1,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[3,0],[0,8]],3,[1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0079","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[1,1],[0,5]]，λ=5，非零v=[1, 4]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"無","explanation_zh":"非零向量殘差=(0, 0)；第一失配=無。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097079,"parameters":{"M":[[1,1],[0,5]],"lambda":5,"v":[1,4],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,5]],5,[1,4]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0080","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[0,5],[1,2]]，λ=0，非零v=[0, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"1","explanation_zh":"非零向量殘差=(5, 2)；第一失配=1。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097080,"parameters":{"M":[[0,5],[1,2]],"lambda":0,"v":[0,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[0,5],[1,2]],0,[0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0081","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[6,0],[0,14]]，λ=6，非零v=[1, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"2","explanation_zh":"非零向量殘差=(0, 8)；第一失配=2。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097081,"parameters":{"M":[[6,0],[0,14]],"lambda":6,"v":[1,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[6,0],[0,14]],6,[1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0082","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[1,1],[0,8]]，λ=8，非零v=[1, 7]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"無","explanation_zh":"非零向量殘差=(0, 0)；第一失配=無。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097082,"parameters":{"M":[[1,1],[0,8]],"lambda":8,"v":[1,7],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,8]],8,[1,7]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0083","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[0,8],[1,2]]，λ=0，非零v=[0, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"1","explanation_zh":"非零向量殘差=(8, 2)；第一失配=1。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097083,"parameters":{"M":[[0,8],[1,2]],"lambda":0,"v":[0,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[0,8],[1,2]],0,[0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0084","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[9,0],[0,20]]，λ=9，非零v=[1, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"2","explanation_zh":"非零向量殘差=(0, 11)；第一失配=2。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097084,"parameters":{"M":[[9,0],[0,20]],"lambda":9,"v":[1,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[9,0],[0,20]],9,[1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0085","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[1,1],[0,11]]，λ=11，非零v=[1, 10]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"無","explanation_zh":"非零向量殘差=(0, 0)；第一失配=無。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097085,"parameters":{"M":[[1,1],[0,11]],"lambda":11,"v":[1,10],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,11]],11,[1,10]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0086","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[0,11],[1,2]]，λ=0，非零v=[0, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"1","explanation_zh":"非零向量殘差=(11, 2)；第一失配=1。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097086,"parameters":{"M":[[0,11],[1,2]],"lambda":0,"v":[0,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[0,11],[1,2]],0,[0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0087","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[12,0],[0,26]]，λ=12，非零v=[1, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"2","explanation_zh":"非零向量殘差=(0, 14)；第一失配=2。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097087,"parameters":{"M":[[12,0],[0,26]],"lambda":12,"v":[1,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[12,0],[0,26]],12,[1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0088","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[1,1],[0,14]]，λ=14，非零v=[1, 13]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"無","explanation_zh":"非零向量殘差=(0, 0)；第一失配=無。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097088,"parameters":{"M":[[1,1],[0,14]],"lambda":14,"v":[1,13],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,14]],14,[1,13]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0089","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[0,14],[1,2]]，λ=0，非零v=[0, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"1","explanation_zh":"非零向量殘差=(14, 2)；第一失配=1。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097089,"parameters":{"M":[[0,14],[1,2]],"lambda":0,"v":[0,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[0,14],[1,2]],0,[0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0090","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[15,0],[0,32]]，λ=15，非零v=[1, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"2","explanation_zh":"非零向量殘差=(0, 17)；第一失配=2。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097090,"parameters":{"M":[[15,0],[0,32]],"lambda":15,"v":[1,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[15,0],[0,32]],15,[1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0091","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[1,1],[0,17]]，λ=17，非零v=[1, 16]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"無","explanation_zh":"非零向量殘差=(0, 0)；第一失配=無。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097091,"parameters":{"M":[[1,1],[0,17]],"lambda":17,"v":[1,16],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,17]],17,[1,16]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0092","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[0,17],[1,2]]，λ=0，非零v=[0, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"1","explanation_zh":"非零向量殘差=(17, 2)；第一失配=1。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097092,"parameters":{"M":[[0,17],[1,2]],"lambda":0,"v":[0,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[0,17],[1,2]],0,[0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0093","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[18,0],[0,38]]，λ=18，非零v=[1, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"2","explanation_zh":"非零向量殘差=(0, 20)；第一失配=2。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097093,"parameters":{"M":[[18,0],[0,38]],"lambda":18,"v":[1,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[18,0],[0,38]],18,[1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0094","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[1,1],[0,20]]，λ=20，非零v=[1, 19]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"無","explanation_zh":"非零向量殘差=(0, 0)；第一失配=無。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097094,"parameters":{"M":[[1,1],[0,20]],"lambda":20,"v":[1,19],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,20]],20,[1,19]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0095","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[0,20],[1,2]]，λ=0，非零v=[0, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"1","explanation_zh":"非零向量殘差=(20, 2)；第一失配=1。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097095,"parameters":{"M":[[0,20],[1,2]],"lambda":0,"v":[0,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[0,20],[1,2]],0,[0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0096","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[21,0],[0,44]]，λ=21，非零v=[1, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"2","explanation_zh":"非零向量殘差=(0, 23)；第一失配=2。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097096,"parameters":{"M":[[21,0],[0,44]],"lambda":21,"v":[1,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[21,0],[0,44]],21,[1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0097","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[1,1],[0,23]]，λ=23，非零v=[1, 22]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"無","explanation_zh":"非零向量殘差=(0, 0)；第一失配=無。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097097,"parameters":{"M":[[1,1],[0,23]],"lambda":23,"v":[1,22],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,23]],23,[1,22]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0098","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[0,23],[1,2]]，λ=0，非零v=[0, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"1","explanation_zh":"非零向量殘差=(23, 2)；第一失配=1。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097098,"parameters":{"M":[[0,23],[1,2]],"lambda":0,"v":[0,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[0,23],[1,2]],0,[0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0099","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[24,0],[0,50]]，λ=24，非零v=[1, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"2","explanation_zh":"非零向量殘差=(0, 26)；第一失配=2。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097099,"parameters":{"M":[[24,0],[0,50]],"lambda":24,"v":[1,1],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[24,0],[0,50]],24,[1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r07-eigenvalue-0100","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零向量候選的第一失配","stem_zh":"實二階方陣與精確整數資料。M=[[1,1],[0,26]]，λ=26，非零v=[1, 25]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。","answer":{"kind":"short_text","value":"無","explanation_zh":"非零向量殘差=(0, 0)；第一失配=無。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零向量候選的第一失配"]},"variant":{"family_id":"qf-density-eigen-first-mismatch","seed":202610097100,"parameters":{"M":[[1,1],[0,26]],"lambda":26,"v":[1,25],"operation":"density_eigen_first_mismatch","domain":"real_two_by_two","semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,26]],26,[1,25]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}