{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0001","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[2, 1, 1], [1, -1, 0]]，有序候選(x,y)=[(-1)/(-3), (1)/(3)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([0/1, 0/1], 是)","explanation_zh":"有限結構核對=([0/1, 0/1], 是)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165001,"parameters":{"rows":[[2,1,1],[1,-1,0]],"candidate":[[-1,-3],[1,3]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[-1,-3],[1,3]],\"rows\":[[2,1,1],[1,-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0002","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 0], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (1)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([0/1, 1/1], 否)","explanation_zh":"有限結構核對=([0/1, 1/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165002,"parameters":{"rows":[[1,0,0],[0,1,0]],"candidate":[[0,1],[1,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[1,1]],\"rows\":[[1,0,0],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0003","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[0, 0, 0], [0, 0, 2]]，有序候選(x,y)=[(0)/(1), (0)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([0/1, -2/1], 否)","explanation_zh":"有限結構核對=([0/1, -2/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165003,"parameters":{"rows":[[0,0,0],[0,0,2]],"candidate":[[0,1],[0,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[0,1]],\"rows\":[[0,0,0],[0,0,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0004","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -9], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (1)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([9/1, 1/1], 否)","explanation_zh":"有限結構核對=([9/1, 1/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165004,"parameters":{"rows":[[1,0,-9],[0,1,0]],"candidate":[[0,1],[1,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[1,1]],\"rows\":[[1,0,-9],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0005","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -8], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (0)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([8/1, 0/1], 否)","explanation_zh":"有限結構核對=([8/1, 0/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165005,"parameters":{"rows":[[1,0,-8],[0,1,0]],"candidate":[[0,1],[0,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[0,1]],\"rows\":[[1,0,-8],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0006","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -7], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (1)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([7/1, 1/1], 否)","explanation_zh":"有限結構核對=([7/1, 1/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165006,"parameters":{"rows":[[1,0,-7],[0,1,0]],"candidate":[[0,1],[1,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[1,1]],\"rows\":[[1,0,-7],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0007","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -6], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (0)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([6/1, 0/1], 否)","explanation_zh":"有限結構核對=([6/1, 0/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165007,"parameters":{"rows":[[1,0,-6],[0,1,0]],"candidate":[[0,1],[0,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[0,1]],\"rows\":[[1,0,-6],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0008","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -5], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (1)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([5/1, 1/1], 否)","explanation_zh":"有限結構核對=([5/1, 1/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165008,"parameters":{"rows":[[1,0,-5],[0,1,0]],"candidate":[[0,1],[1,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[1,1]],\"rows\":[[1,0,-5],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0009","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -4], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (0)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([4/1, 0/1], 否)","explanation_zh":"有限結構核對=([4/1, 0/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165009,"parameters":{"rows":[[1,0,-4],[0,1,0]],"candidate":[[0,1],[0,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[0,1]],\"rows\":[[1,0,-4],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0010","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -3], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (1)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([3/1, 1/1], 否)","explanation_zh":"有限結構核對=([3/1, 1/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165010,"parameters":{"rows":[[1,0,-3],[0,1,0]],"candidate":[[0,1],[1,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[1,1]],\"rows\":[[1,0,-3],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0011","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -2], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (0)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([2/1, 0/1], 否)","explanation_zh":"有限結構核對=([2/1, 0/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165011,"parameters":{"rows":[[1,0,-2],[0,1,0]],"candidate":[[0,1],[0,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[0,1]],\"rows\":[[1,0,-2],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0012","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -1], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (1)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([1/1, 1/1], 否)","explanation_zh":"有限結構核對=([1/1, 1/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165012,"parameters":{"rows":[[1,0,-1],[0,1,0]],"candidate":[[0,1],[1,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[1,1]],\"rows\":[[1,0,-1],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0013","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 0], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (2)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([0/1, 2/1], 否)","explanation_zh":"有限結構核對=([0/1, 2/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165013,"parameters":{"rows":[[1,0,0],[0,1,0]],"candidate":[[0,1],[2,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[2,1]],\"rows\":[[1,0,0],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0014","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 1], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (1)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([-1/1, 1/1], 否)","explanation_zh":"有限結構核對=([-1/1, 1/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165014,"parameters":{"rows":[[1,0,1],[0,1,0]],"candidate":[[0,1],[1,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[1,1]],\"rows\":[[1,0,1],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0015","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 2], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (0)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([-2/1, 0/1], 否)","explanation_zh":"有限結構核對=([-2/1, 0/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165015,"parameters":{"rows":[[1,0,2],[0,1,0]],"candidate":[[0,1],[0,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[0,1]],\"rows\":[[1,0,2],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0016","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 3], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (1)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([-3/1, 1/1], 否)","explanation_zh":"有限結構核對=([-3/1, 1/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165016,"parameters":{"rows":[[1,0,3],[0,1,0]],"candidate":[[0,1],[1,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[1,1]],\"rows\":[[1,0,3],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0017","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 4], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (0)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([-4/1, 0/1], 否)","explanation_zh":"有限結構核對=([-4/1, 0/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165017,"parameters":{"rows":[[1,0,4],[0,1,0]],"candidate":[[0,1],[0,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[0,1]],\"rows\":[[1,0,4],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0018","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 5], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (1)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([-5/1, 1/1], 否)","explanation_zh":"有限結構核對=([-5/1, 1/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165018,"parameters":{"rows":[[1,0,5],[0,1,0]],"candidate":[[0,1],[1,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[1,1]],\"rows\":[[1,0,5],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0019","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 6], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (0)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([-6/1, 0/1], 否)","explanation_zh":"有限結構核對=([-6/1, 0/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165019,"parameters":{"rows":[[1,0,6],[0,1,0]],"candidate":[[0,1],[0,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[0,1]],\"rows\":[[1,0,6],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0020","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 7], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (1)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([-7/1, 1/1], 否)","explanation_zh":"有限結構核對=([-7/1, 1/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165020,"parameters":{"rows":[[1,0,7],[0,1,0]],"candidate":[[0,1],[1,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[1,1]],\"rows\":[[1,0,7],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0021","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 8], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (0)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([-8/1, 0/1], 否)","explanation_zh":"有限結構核對=([-8/1, 0/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165021,"parameters":{"rows":[[1,0,8],[0,1,0]],"candidate":[[0,1],[0,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[0,1]],\"rows\":[[1,0,8],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0022","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 9], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (1)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([-9/1, 1/1], 否)","explanation_zh":"有限結構核對=([-9/1, 1/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165022,"parameters":{"rows":[[1,0,9],[0,1,0]],"candidate":[[0,1],[1,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[1,1]],\"rows\":[[1,0,9],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0023","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 10], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (0)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([-10/1, 0/1], 否)","explanation_zh":"有限結構核對=([-10/1, 0/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165023,"parameters":{"rows":[[1,0,10],[0,1,0]],"candidate":[[0,1],[0,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[0,1]],\"rows\":[[1,0,10],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0024","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 11], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (1)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([-11/1, 1/1], 否)","explanation_zh":"有限結構核對=([-11/1, 1/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165024,"parameters":{"rows":[[1,0,11],[0,1,0]],"candidate":[[0,1],[1,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[1,1]],\"rows\":[[1,0,11],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0025","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"候選同時滿足兩式的殘差","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 12], [0, 1, 0]]，有序候選(x,y)=[(0)/(1), (0)/(1)]。求 (兩列a*x+b*y−c的有理殘差, 兩列同時零資格)。","answer":{"kind":"expression","value":"([-12/1, 0/1], 否)","explanation_zh":"有限結構核對=([-12/1, 0/1], 否)。先規約合法候選分數，再分別代入兩列；只過一列不是系統的解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選同時滿足兩式的殘差"]},"variant":{"family_id":"qf-density16-system-candidate-two-residuals","seed":202610165025,"parameters":{"rows":[[1,0,12],[0,1,0]],"candidate":[[0,1],[0,1]],"domain":"R2_linear_system_integer_slice","operation":"system-candidate-two-residuals","semantic_template":"qf-density16-system-candidate-two-residuals/{\"candidate\":[[0,1],[0,1]],\"rows\":[[1,0,12],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0026","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[0, 0, 0], [0, 0, 0]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(0, 0, 0, 無限多解, 不唯一)","explanation_zh":"有限結構核對=(0, 0, 0, 無限多解, 不唯一)。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165026,"parameters":{"rows":[[0,0,0],[0,0,0]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[0,0,0],[0,0,0]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0027","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[0, 0, 1], [0, 0, 0]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(0, 0, 1, 無解, 不適用)","explanation_zh":"有限結構核對=(0, 0, 1, 無解, 不適用)。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165027,"parameters":{"rows":[[0,0,1],[0,0,0]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[0,0,1],[0,0,0]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0028","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 1], [2, 2, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(0, 1, 1, 無限多解, 不唯一)","explanation_zh":"有限結構核對=(0, 1, 1, 無限多解, 不唯一)。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165028,"parameters":{"rows":[[1,1,1],[2,2,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,1],[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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0029","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 1], [2, 2, 3]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(0, 1, 2, 無解, 不適用)","explanation_zh":"有限結構核對=(0, 1, 2, 無解, 不適用)。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165029,"parameters":{"rows":[[1,1,1],[2,2,3]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,1],[2,2,3]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0030","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[2, 1, 1], [1, -1, 0]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(-3, 2, 2, 唯一解, [1/3, 1/3])","explanation_zh":"有限結構核對=(-3, 2, 2, 唯一解, [1/3, 1/3])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165030,"parameters":{"rows":[[2,1,1],[1,-1,0]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[2,1,1],[1,-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0031","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, -7], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [-9/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [-9/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165031,"parameters":{"rows":[[1,1,-7],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,-7],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0032","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, -6], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [-8/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [-8/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165032,"parameters":{"rows":[[1,1,-6],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,-6],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0033","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, -5], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [-7/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [-7/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165033,"parameters":{"rows":[[1,1,-5],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,-5],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0034","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, -4], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [-6/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [-6/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165034,"parameters":{"rows":[[1,1,-4],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,-4],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0035","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, -3], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [-5/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [-5/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165035,"parameters":{"rows":[[1,1,-3],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,-3],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0036","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, -2], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [-4/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [-4/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165036,"parameters":{"rows":[[1,1,-2],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,-2],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0037","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, -1], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [-3/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [-3/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165037,"parameters":{"rows":[[1,1,-1],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,-1],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0038","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 0], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [-2/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [-2/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165038,"parameters":{"rows":[[1,1,0],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,0],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0039","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 1], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [-1/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [-1/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165039,"parameters":{"rows":[[1,1,1],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,1],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0040","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 2], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [0/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [0/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165040,"parameters":{"rows":[[1,1,2],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,2],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0041","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 3], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [1/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [1/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165041,"parameters":{"rows":[[1,1,3],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,3],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0042","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 4], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [2/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [2/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165042,"parameters":{"rows":[[1,1,4],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,4],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0043","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 5], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [3/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [3/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165043,"parameters":{"rows":[[1,1,5],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,5],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0044","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 6], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [4/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [4/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165044,"parameters":{"rows":[[1,1,6],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,6],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0045","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 7], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [5/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [5/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165045,"parameters":{"rows":[[1,1,7],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,7],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0046","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 8], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [6/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [6/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165046,"parameters":{"rows":[[1,1,8],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,8],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0047","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 9], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [7/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [7/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165047,"parameters":{"rows":[[1,1,9],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,9],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0048","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 10], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [8/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [8/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165048,"parameters":{"rows":[[1,1,10],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,10],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0049","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 11], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [9/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [9/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165049,"parameters":{"rows":[[1,1,11],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,11],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0050","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"兩列秩與完整解分類","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 12], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。","answer":{"kind":"expression","value":"(1, 2, 2, 唯一解, [10/1, 2/1])","explanation_zh":"有限結構核對=(1, 2, 2, 唯一解, [10/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩列秩與完整解分類"]},"variant":{"family_id":"qf-density16-system-classify-solve","seed":202610165050,"parameters":{"rows":[[1,1,12],[0,1,2]],"domain":"R2_linear_system_integer_slice","operation":"system-classify-solve","semantic_template":"qf-density16-system-classify-solve/{\"rows\":[[1,1,12],[0,1,2]]}"}},"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0051","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[0, 0, 0], [0, 0, 0]]、t=0。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[0, 0, 0], [0, 0, 0]], [0, 0, 0, 無限多解, 不唯一], [0, 0, 0, 無限多解, 不唯一], 是)","explanation_zh":"有限結構核對=([[0, 0, 0], [0, 0, 0]], [0, 0, 0, 無限多解, 不唯一], [0, 0, 0, 無限多解, 不唯一], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165051,"parameters":{"rows":[[0,0,0],[0,0,0]],"t":0,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[0,0,0],[0,0,0]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0052","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 1], [2, 2, 2]]、t=-2。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 1, 1], [0, 0, 0]], [0, 1, 1, 無限多解, 不唯一], [0, 1, 1, 無限多解, 不唯一], 是)","explanation_zh":"有限結構核對=([[1, 1, 1], [0, 0, 0]], [0, 1, 1, 無限多解, 不唯一], [0, 1, 1, 無限多解, 不唯一], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165052,"parameters":{"rows":[[1,1,1],[2,2,2]],"t":-2,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,1,1],[2,2,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0053","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 1], [2, 2, 3]]、t=-2。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 1, 1], [0, 0, 1]], [0, 1, 2, 無解, 不適用], [0, 1, 2, 無解, 不適用], 是)","explanation_zh":"有限結構核對=([[1, 1, 1], [0, 0, 1]], [0, 1, 2, 無解, 不適用], [0, 1, 2, 無解, 不適用], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165053,"parameters":{"rows":[[1,1,1],[2,2,3]],"t":-2,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,1,1],[2,2,3]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0054","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[2, 1, 1], [1, -1, 0]]、t=-1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[2, 1, 1], [-1, -2, -1]], [-3, 2, 2, 唯一解, [1/3, 1/3]], [-3, 2, 2, 唯一解, [1/3, 1/3]], 是)","explanation_zh":"有限結構核對=([[2, 1, 1], [-1, -2, -1]], [-3, 2, 2, 唯一解, [1/3, 1/3]], [-3, 2, 2, 唯一解, [1/3, 1/3]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165054,"parameters":{"rows":[[2,1,1],[1,-1,0]],"t":-1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[2,1,1],[1,-1,0]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0055","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -8], [0, 1, 2]]、t=0。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, -8], [0, 1, 2]], [1, 2, 2, 唯一解, [-8/1, 2/1]], [1, 2, 2, 唯一解, [-8/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, -8], [0, 1, 2]], [1, 2, 2, 唯一解, [-8/1, 2/1]], [1, 2, 2, 唯一解, [-8/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165055,"parameters":{"rows":[[1,0,-8],[0,1,2]],"t":0,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,-8],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0056","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -7], [0, 1, 2]]、t=1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, -7], [1, 1, -5]], [1, 2, 2, 唯一解, [-7/1, 2/1]], [1, 2, 2, 唯一解, [-7/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, -7], [1, 1, -5]], [1, 2, 2, 唯一解, [-7/1, 2/1]], [1, 2, 2, 唯一解, [-7/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165056,"parameters":{"rows":[[1,0,-7],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,-7],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0057","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -6], [0, 1, 2]]、t=-1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, -6], [-1, 1, 8]], [1, 2, 2, 唯一解, [-6/1, 2/1]], [1, 2, 2, 唯一解, [-6/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, -6], [-1, 1, 8]], [1, 2, 2, 唯一解, [-6/1, 2/1]], [1, 2, 2, 唯一解, [-6/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165057,"parameters":{"rows":[[1,0,-6],[0,1,2]],"t":-1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,-6],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0058","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -5], [0, 1, 2]]、t=0。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, -5], [0, 1, 2]], [1, 2, 2, 唯一解, [-5/1, 2/1]], [1, 2, 2, 唯一解, [-5/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, -5], [0, 1, 2]], [1, 2, 2, 唯一解, [-5/1, 2/1]], [1, 2, 2, 唯一解, [-5/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165058,"parameters":{"rows":[[1,0,-5],[0,1,2]],"t":0,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,-5],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0059","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -4], [0, 1, 2]]、t=1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, -4], [1, 1, -2]], [1, 2, 2, 唯一解, [-4/1, 2/1]], [1, 2, 2, 唯一解, [-4/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, -4], [1, 1, -2]], [1, 2, 2, 唯一解, [-4/1, 2/1]], [1, 2, 2, 唯一解, [-4/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165059,"parameters":{"rows":[[1,0,-4],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,-4],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0060","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -3], [0, 1, 2]]、t=-1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, -3], [-1, 1, 5]], [1, 2, 2, 唯一解, [-3/1, 2/1]], [1, 2, 2, 唯一解, [-3/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, -3], [-1, 1, 5]], [1, 2, 2, 唯一解, [-3/1, 2/1]], [1, 2, 2, 唯一解, [-3/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165060,"parameters":{"rows":[[1,0,-3],[0,1,2]],"t":-1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,-3],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0061","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -2], [0, 1, 2]]、t=0。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, -2], [0, 1, 2]], [1, 2, 2, 唯一解, [-2/1, 2/1]], [1, 2, 2, 唯一解, [-2/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, -2], [0, 1, 2]], [1, 2, 2, 唯一解, [-2/1, 2/1]], [1, 2, 2, 唯一解, [-2/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165061,"parameters":{"rows":[[1,0,-2],[0,1,2]],"t":0,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,-2],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0062","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -1], [0, 1, 2]]、t=1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, -1], [1, 1, 1]], [1, 2, 2, 唯一解, [-1/1, 2/1]], [1, 2, 2, 唯一解, [-1/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, -1], [1, 1, 1]], [1, 2, 2, 唯一解, [-1/1, 2/1]], [1, 2, 2, 唯一解, [-1/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165062,"parameters":{"rows":[[1,0,-1],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,-1],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0063","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 0], [0, 1, 2]]、t=-1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, 0], [-1, 1, 2]], [1, 2, 2, 唯一解, [0/1, 2/1]], [1, 2, 2, 唯一解, [0/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, 0], [-1, 1, 2]], [1, 2, 2, 唯一解, [0/1, 2/1]], [1, 2, 2, 唯一解, [0/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165063,"parameters":{"rows":[[1,0,0],[0,1,2]],"t":-1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,0],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0064","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 1], [0, 1, 2]]、t=0。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, 1], [0, 1, 2]], [1, 2, 2, 唯一解, [1/1, 2/1]], [1, 2, 2, 唯一解, [1/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, 1], [0, 1, 2]], [1, 2, 2, 唯一解, [1/1, 2/1]], [1, 2, 2, 唯一解, [1/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165064,"parameters":{"rows":[[1,0,1],[0,1,2]],"t":0,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,1],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0065","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 2], [0, 1, 2]]、t=1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, 2], [1, 1, 4]], [1, 2, 2, 唯一解, [2/1, 2/1]], [1, 2, 2, 唯一解, [2/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, 2], [1, 1, 4]], [1, 2, 2, 唯一解, [2/1, 2/1]], [1, 2, 2, 唯一解, [2/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165065,"parameters":{"rows":[[1,0,2],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,2],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0066","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 3], [0, 1, 2]]、t=-1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, 3], [-1, 1, -1]], [1, 2, 2, 唯一解, [3/1, 2/1]], [1, 2, 2, 唯一解, [3/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, 3], [-1, 1, -1]], [1, 2, 2, 唯一解, [3/1, 2/1]], [1, 2, 2, 唯一解, [3/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165066,"parameters":{"rows":[[1,0,3],[0,1,2]],"t":-1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,3],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0067","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 4], [0, 1, 2]]、t=0。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, 4], [0, 1, 2]], [1, 2, 2, 唯一解, [4/1, 2/1]], [1, 2, 2, 唯一解, [4/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, 4], [0, 1, 2]], [1, 2, 2, 唯一解, [4/1, 2/1]], [1, 2, 2, 唯一解, [4/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165067,"parameters":{"rows":[[1,0,4],[0,1,2]],"t":0,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,4],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0068","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 5], [0, 1, 2]]、t=1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, 5], [1, 1, 7]], [1, 2, 2, 唯一解, [5/1, 2/1]], [1, 2, 2, 唯一解, [5/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, 5], [1, 1, 7]], [1, 2, 2, 唯一解, [5/1, 2/1]], [1, 2, 2, 唯一解, [5/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165068,"parameters":{"rows":[[1,0,5],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,5],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0069","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 6], [0, 1, 2]]、t=-1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, 6], [-1, 1, -4]], [1, 2, 2, 唯一解, [6/1, 2/1]], [1, 2, 2, 唯一解, [6/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, 6], [-1, 1, -4]], [1, 2, 2, 唯一解, [6/1, 2/1]], [1, 2, 2, 唯一解, [6/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165069,"parameters":{"rows":[[1,0,6],[0,1,2]],"t":-1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,6],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0070","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 7], [0, 1, 2]]、t=0。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, 7], [0, 1, 2]], [1, 2, 2, 唯一解, [7/1, 2/1]], [1, 2, 2, 唯一解, [7/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, 7], [0, 1, 2]], [1, 2, 2, 唯一解, [7/1, 2/1]], [1, 2, 2, 唯一解, [7/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165070,"parameters":{"rows":[[1,0,7],[0,1,2]],"t":0,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,7],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0071","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 8], [0, 1, 2]]、t=1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, 8], [1, 1, 10]], [1, 2, 2, 唯一解, [8/1, 2/1]], [1, 2, 2, 唯一解, [8/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, 8], [1, 1, 10]], [1, 2, 2, 唯一解, [8/1, 2/1]], [1, 2, 2, 唯一解, [8/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165071,"parameters":{"rows":[[1,0,8],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,8],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0072","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 9], [0, 1, 2]]、t=-1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, 9], [-1, 1, -7]], [1, 2, 2, 唯一解, [9/1, 2/1]], [1, 2, 2, 唯一解, [9/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, 9], [-1, 1, -7]], [1, 2, 2, 唯一解, [9/1, 2/1]], [1, 2, 2, 唯一解, [9/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165072,"parameters":{"rows":[[1,0,9],[0,1,2]],"t":-1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,9],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0073","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 10], [0, 1, 2]]、t=0。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, 10], [0, 1, 2]], [1, 2, 2, 唯一解, [10/1, 2/1]], [1, 2, 2, 唯一解, [10/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, 10], [0, 1, 2]], [1, 2, 2, 唯一解, [10/1, 2/1]], [1, 2, 2, 唯一解, [10/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165073,"parameters":{"rows":[[1,0,10],[0,1,2]],"t":0,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,10],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0074","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 11], [0, 1, 2]]、t=1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, 11], [1, 1, 13]], [1, 2, 2, 唯一解, [11/1, 2/1]], [1, 2, 2, 唯一解, [11/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, 11], [1, 1, 13]], [1, 2, 2, 唯一解, [11/1, 2/1]], [1, 2, 2, 唯一解, [11/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165074,"parameters":{"rows":[[1,0,11],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,11],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0075","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"加倍第一列的消去操作","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 12], [0, 1, 2]]、t=-1。只把row2改row2+t*row1。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, 可逆列操作資格)。","answer":{"kind":"expression","value":"([[1, 0, 12], [-1, 1, -10]], [1, 2, 2, 唯一解, [12/1, 2/1]], [1, 2, 2, 唯一解, [12/1, 2/1]], 是)","explanation_zh":"有限結構核對=([[1, 0, 12], [-1, 1, -10]], [1, 2, 2, 唯一解, [12/1, 2/1]], [1, 2, 2, 唯一解, [12/1, 2/1]], 是)。整列連右側常數一起相加，再用Gaussian分別核前後；反操作為加負t倍第一列。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加倍第一列的消去操作"]},"variant":{"family_id":"qf-density16-system-eliminate-row","seed":202610165075,"parameters":{"rows":[[1,0,12],[0,1,2]],"t":-1,"domain":"R2_linear_system_integer_slice","operation":"system-eliminate-row","semantic_template":"qf-density16-system-eliminate-row/{\"rows\":[[1,0,12],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0076","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[0, 0, 0], [0, 0, 0]]、t=0。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[0, 0, 0], [0, 0, 0]], [0, 0, 0, 無限多解, 不唯一], [0, 0, 0, 無限多解, 不唯一], 否, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[0, 0, 0], [0, 0, 0]], [0, 0, 0, 無限多解, 不唯一], [0, 0, 0, 無限多解, 不唯一], 否, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165076,"parameters":{"rows":[[0,0,0],[0,0,0]],"t":0,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[0,0,0],[0,0,0]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0077","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 1], [2, 2, 2]]、t=0。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[0, 0, 0], [2, 2, 2]], [0, 1, 1, 無限多解, 不唯一], [0, 1, 1, 無限多解, 不唯一], 否, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[0, 0, 0], [2, 2, 2]], [0, 1, 1, 無限多解, 不唯一], [0, 1, 1, 無限多解, 不唯一], 否, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165077,"parameters":{"rows":[[1,1,1],[2,2,2]],"t":0,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,1,1],[2,2,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0078","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 1], [0, 1, 2]]、t=0。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[0, 0, 0], [0, 1, 2]], [1, 2, 2, 唯一解, [1/1, 2/1]], [0, 1, 1, 無限多解, 不唯一], 否, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[0, 0, 0], [0, 1, 2]], [1, 2, 2, 唯一解, [1/1, 2/1]], [0, 1, 1, 無限多解, 不唯一], 否, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165078,"parameters":{"rows":[[1,0,1],[0,1,2]],"t":0,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,1],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0079","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 1], [2, 2, 3]]、t=0。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[0, 0, 0], [2, 2, 3]], [0, 1, 2, 無解, 不適用], [0, 1, 1, 無限多解, 不唯一], 否, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[0, 0, 0], [2, 2, 3]], [0, 1, 2, 無解, 不適用], [0, 1, 1, 無限多解, 不唯一], 否, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165079,"parameters":{"rows":[[1,1,1],[2,2,3]],"t":0,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,1,1],[2,2,3]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0080","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -8], [0, 1, 2]]、t=1。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[1, 0, -8], [0, 1, 2]], [1, 2, 2, 唯一解, [-8/1, 2/1]], [1, 2, 2, 唯一解, [-8/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[1, 0, -8], [0, 1, 2]], [1, 2, 2, 唯一解, [-8/1, 2/1]], [1, 2, 2, 唯一解, [-8/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165080,"parameters":{"rows":[[1,0,-8],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,-8],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0081","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -7], [0, 1, 2]]、t=-2。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[-2, 0, 14], [0, 1, 2]], [1, 2, 2, 唯一解, [-7/1, 2/1]], [-2, 2, 2, 唯一解, [-7/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[-2, 0, 14], [0, 1, 2]], [1, 2, 2, 唯一解, [-7/1, 2/1]], [-2, 2, 2, 唯一解, [-7/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165081,"parameters":{"rows":[[1,0,-7],[0,1,2]],"t":-2,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,-7],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0082","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -6], [0, 1, 2]]、t=1。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[1, 0, -6], [0, 1, 2]], [1, 2, 2, 唯一解, [-6/1, 2/1]], [1, 2, 2, 唯一解, [-6/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[1, 0, -6], [0, 1, 2]], [1, 2, 2, 唯一解, [-6/1, 2/1]], [1, 2, 2, 唯一解, [-6/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165082,"parameters":{"rows":[[1,0,-6],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,-6],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0083","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -5], [0, 1, 2]]、t=-2。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[-2, 0, 10], [0, 1, 2]], [1, 2, 2, 唯一解, [-5/1, 2/1]], [-2, 2, 2, 唯一解, [-5/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[-2, 0, 10], [0, 1, 2]], [1, 2, 2, 唯一解, [-5/1, 2/1]], [-2, 2, 2, 唯一解, [-5/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165083,"parameters":{"rows":[[1,0,-5],[0,1,2]],"t":-2,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,-5],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0084","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -4], [0, 1, 2]]、t=1。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[1, 0, -4], [0, 1, 2]], [1, 2, 2, 唯一解, [-4/1, 2/1]], [1, 2, 2, 唯一解, [-4/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[1, 0, -4], [0, 1, 2]], [1, 2, 2, 唯一解, [-4/1, 2/1]], [1, 2, 2, 唯一解, [-4/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165084,"parameters":{"rows":[[1,0,-4],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,-4],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0085","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -3], [0, 1, 2]]、t=-2。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[-2, 0, 6], [0, 1, 2]], [1, 2, 2, 唯一解, [-3/1, 2/1]], [-2, 2, 2, 唯一解, [-3/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[-2, 0, 6], [0, 1, 2]], [1, 2, 2, 唯一解, [-3/1, 2/1]], [-2, 2, 2, 唯一解, [-3/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165085,"parameters":{"rows":[[1,0,-3],[0,1,2]],"t":-2,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,-3],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0086","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -2], [0, 1, 2]]、t=1。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[1, 0, -2], [0, 1, 2]], [1, 2, 2, 唯一解, [-2/1, 2/1]], [1, 2, 2, 唯一解, [-2/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[1, 0, -2], [0, 1, 2]], [1, 2, 2, 唯一解, [-2/1, 2/1]], [1, 2, 2, 唯一解, [-2/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165086,"parameters":{"rows":[[1,0,-2],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,-2],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0087","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, -1], [0, 1, 2]]、t=-2。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[-2, 0, 2], [0, 1, 2]], [1, 2, 2, 唯一解, [-1/1, 2/1]], [-2, 2, 2, 唯一解, [-1/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[-2, 0, 2], [0, 1, 2]], [1, 2, 2, 唯一解, [-1/1, 2/1]], [-2, 2, 2, 唯一解, [-1/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165087,"parameters":{"rows":[[1,0,-1],[0,1,2]],"t":-2,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,-1],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0088","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 0], [0, 1, 2]]、t=1。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[1, 0, 0], [0, 1, 2]], [1, 2, 2, 唯一解, [0/1, 2/1]], [1, 2, 2, 唯一解, [0/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[1, 0, 0], [0, 1, 2]], [1, 2, 2, 唯一解, [0/1, 2/1]], [1, 2, 2, 唯一解, [0/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165088,"parameters":{"rows":[[1,0,0],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,0],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0089","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 1], [0, 1, 2]]、t=-2。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[-2, 0, -2], [0, 1, 2]], [1, 2, 2, 唯一解, [1/1, 2/1]], [-2, 2, 2, 唯一解, [1/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[-2, 0, -2], [0, 1, 2]], [1, 2, 2, 唯一解, [1/1, 2/1]], [-2, 2, 2, 唯一解, [1/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165089,"parameters":{"rows":[[1,0,1],[0,1,2]],"t":-2,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,1],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0090","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 2], [0, 1, 2]]、t=1。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[1, 0, 2], [0, 1, 2]], [1, 2, 2, 唯一解, [2/1, 2/1]], [1, 2, 2, 唯一解, [2/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[1, 0, 2], [0, 1, 2]], [1, 2, 2, 唯一解, [2/1, 2/1]], [1, 2, 2, 唯一解, [2/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165090,"parameters":{"rows":[[1,0,2],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,2],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0091","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 3], [0, 1, 2]]、t=-2。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[-2, 0, -6], [0, 1, 2]], [1, 2, 2, 唯一解, [3/1, 2/1]], [-2, 2, 2, 唯一解, [3/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[-2, 0, -6], [0, 1, 2]], [1, 2, 2, 唯一解, [3/1, 2/1]], [-2, 2, 2, 唯一解, [3/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165091,"parameters":{"rows":[[1,0,3],[0,1,2]],"t":-2,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,3],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0092","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 4], [0, 1, 2]]、t=1。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[1, 0, 4], [0, 1, 2]], [1, 2, 2, 唯一解, [4/1, 2/1]], [1, 2, 2, 唯一解, [4/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[1, 0, 4], [0, 1, 2]], [1, 2, 2, 唯一解, [4/1, 2/1]], [1, 2, 2, 唯一解, [4/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165092,"parameters":{"rows":[[1,0,4],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,4],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0093","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 5], [0, 1, 2]]、t=-2。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[-2, 0, -10], [0, 1, 2]], [1, 2, 2, 唯一解, [5/1, 2/1]], [-2, 2, 2, 唯一解, [5/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[-2, 0, -10], [0, 1, 2]], [1, 2, 2, 唯一解, [5/1, 2/1]], [-2, 2, 2, 唯一解, [5/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165093,"parameters":{"rows":[[1,0,5],[0,1,2]],"t":-2,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,5],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0094","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 6], [0, 1, 2]]、t=1。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[1, 0, 6], [0, 1, 2]], [1, 2, 2, 唯一解, [6/1, 2/1]], [1, 2, 2, 唯一解, [6/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[1, 0, 6], [0, 1, 2]], [1, 2, 2, 唯一解, [6/1, 2/1]], [1, 2, 2, 唯一解, [6/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165094,"parameters":{"rows":[[1,0,6],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,6],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0095","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 7], [0, 1, 2]]、t=-2。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[-2, 0, -14], [0, 1, 2]], [1, 2, 2, 唯一解, [7/1, 2/1]], [-2, 2, 2, 唯一解, [7/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[-2, 0, -14], [0, 1, 2]], [1, 2, 2, 唯一解, [7/1, 2/1]], [-2, 2, 2, 唯一解, [7/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165095,"parameters":{"rows":[[1,0,7],[0,1,2]],"t":-2,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,7],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0096","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 8], [0, 1, 2]]、t=1。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[1, 0, 8], [0, 1, 2]], [1, 2, 2, 唯一解, [8/1, 2/1]], [1, 2, 2, 唯一解, [8/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[1, 0, 8], [0, 1, 2]], [1, 2, 2, 唯一解, [8/1, 2/1]], [1, 2, 2, 唯一解, [8/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165096,"parameters":{"rows":[[1,0,8],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,8],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0097","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 9], [0, 1, 2]]、t=-2。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[-2, 0, -18], [0, 1, 2]], [1, 2, 2, 唯一解, [9/1, 2/1]], [-2, 2, 2, 唯一解, [9/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[-2, 0, -18], [0, 1, 2]], [1, 2, 2, 唯一解, [9/1, 2/1]], [-2, 2, 2, 唯一解, [9/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165097,"parameters":{"rows":[[1,0,9],[0,1,2]],"t":-2,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,9],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0098","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 10], [0, 1, 2]]、t=1。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[1, 0, 10], [0, 1, 2]], [1, 2, 2, 唯一解, [10/1, 2/1]], [1, 2, 2, 唯一解, [10/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[1, 0, 10], [0, 1, 2]], [1, 2, 2, 唯一解, [10/1, 2/1]], [1, 2, 2, 唯一解, [10/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165098,"parameters":{"rows":[[1,0,10],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,10],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0099","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 11], [0, 1, 2]]、t=-2。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[-2, 0, -22], [0, 1, 2]], [1, 2, 2, 唯一解, [11/1, 2/1]], [-2, 2, 2, 唯一解, [11/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[-2, 0, -22], [0, 1, 2]], [1, 2, 2, 唯一解, [11/1, 2/1]], [-2, 2, 2, 唯一解, [11/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165099,"parameters":{"rows":[[1,0,11],[0,1,2]],"t":-2,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,11],[0,1,2]],\"t\":-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r05-system-linear-equations-0100","batch_id":"batch-20261012-advance-20261016-r05-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"第一列縮放的操作資格","stem_zh":"R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 12], [0, 1, 2]]、t=1。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。","answer":{"kind":"expression","value":"([[1, 0, 12], [0, 1, 2]], [1, 2, 2, 唯一解, [12/1, 2/1]], [1, 2, 2, 唯一解, [12/1, 2/1]], 是, 非零縮放才是可逆列操作)","explanation_zh":"有限結構核對=([[1, 0, 12], [0, 1, 2]], [1, 2, 2, 唯一解, [12/1, 2/1]], [1, 2, 2, 唯一解, [12/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["第一列縮放的操作資格"]},"variant":{"family_id":"qf-density16-system-row-scaling-solution","seed":202610165100,"parameters":{"rows":[[1,0,12],[0,1,2]],"t":1,"domain":"R2_linear_system_integer_slice","operation":"system-row-scaling-solution","semantic_template":"qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,12],[0,1,2]],\"t\":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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}