{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0001","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (0)x+(0)y=0 與 (0)x+(0)y=0 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"D=0。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046001,"parameters":{"rows":[[0,0,0],[0,0,0]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"0\",\"0\"],[\"0\",\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0002","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (-11)x+(1)y=-1 與 (1)x+(-1)y=-2 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"10","explanation_zh":"D=10。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046002,"parameters":{"rows":[[-11,1,-1],[1,-1,-2]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-11\",\"1\"],[\"1\",\"-1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0003","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (-10)x+(1)y=0 與 (1)x+(0)y=-1 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"D=-1。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046003,"parameters":{"rows":[[-10,1,0],[1,0,-1]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-10\",\"1\"],[\"1\",\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0004","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (-9)x+(1)y=1 與 (1)x+(1)y=0 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"-10","explanation_zh":"D=-10。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046004,"parameters":{"rows":[[-9,1,1],[1,1,0]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-9\",\"1\"],[\"1\",\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0005","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (-8)x+(1)y=2 與 (1)x+(2)y=1 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"-17","explanation_zh":"D=-17。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046005,"parameters":{"rows":[[-8,1,2],[1,2,1]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-8\",\"1\"],[\"1\",\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0006","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (-7)x+(1)y=-2 與 (1)x+(-2)y=2 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"13","explanation_zh":"D=13。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046006,"parameters":{"rows":[[-7,1,-2],[1,-2,2]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-7\",\"1\"],[\"1\",\"-2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0007","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (-6)x+(1)y=-1 與 (1)x+(-1)y=3 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"5","explanation_zh":"D=5。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046007,"parameters":{"rows":[[-6,1,-1],[1,-1,3]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-6\",\"1\"],[\"1\",\"-1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0008","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (-5)x+(1)y=0 與 (1)x+(0)y=-3 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"D=-1。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046008,"parameters":{"rows":[[-5,1,0],[1,0,-3]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-5\",\"1\"],[\"1\",\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0009","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (-4)x+(1)y=1 與 (1)x+(1)y=-2 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"-5","explanation_zh":"D=-5。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046009,"parameters":{"rows":[[-4,1,1],[1,1,-2]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-4\",\"1\"],[\"1\",\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0010","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (-3)x+(1)y=2 與 (1)x+(2)y=-1 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"-7","explanation_zh":"D=-7。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046010,"parameters":{"rows":[[-3,1,2],[1,2,-1]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-3\",\"1\"],[\"1\",\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0011","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (-2)x+(1)y=-2 與 (1)x+(-2)y=0 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"3","explanation_zh":"D=3。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046011,"parameters":{"rows":[[-2,1,-2],[1,-2,0]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-2\",\"1\"],[\"1\",\"-2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0012","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (-1)x+(1)y=-1 與 (1)x+(-1)y=1 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"D=0。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046012,"parameters":{"rows":[[-1,1,-1],[1,-1,1]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-1\",\"1\"],[\"1\",\"-1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0013","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (0)x+(1)y=0 與 (1)x+(0)y=2 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"D=-1。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046013,"parameters":{"rows":[[0,1,0],[1,0,2]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"0\",\"1\"],[\"1\",\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0014","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (1)x+(1)y=1 與 (1)x+(1)y=3 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"D=0。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046014,"parameters":{"rows":[[1,1,1],[1,1,3]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"1\",\"1\"],[\"1\",\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0015","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (2)x+(1)y=2 與 (1)x+(2)y=-3 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"3","explanation_zh":"D=3。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046015,"parameters":{"rows":[[2,1,2],[1,2,-3]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"2\",\"1\"],[\"1\",\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0016","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (3)x+(1)y=-2 與 (1)x+(-2)y=-2 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"-7","explanation_zh":"D=-7。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046016,"parameters":{"rows":[[3,1,-2],[1,-2,-2]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"3\",\"1\"],[\"1\",\"-2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0017","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (4)x+(1)y=-1 與 (1)x+(-1)y=-1 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"-5","explanation_zh":"D=-5。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046017,"parameters":{"rows":[[4,1,-1],[1,-1,-1]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"4\",\"1\"],[\"1\",\"-1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0018","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (5)x+(1)y=0 與 (1)x+(0)y=0 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"D=-1。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046018,"parameters":{"rows":[[5,1,0],[1,0,0]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"5\",\"1\"],[\"1\",\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0019","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (6)x+(1)y=1 與 (1)x+(1)y=1 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"5","explanation_zh":"D=5。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046019,"parameters":{"rows":[[6,1,1],[1,1,1]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"6\",\"1\"],[\"1\",\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0020","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (7)x+(1)y=2 與 (1)x+(2)y=2 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"13","explanation_zh":"D=13。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046020,"parameters":{"rows":[[7,1,2],[1,2,2]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"7\",\"1\"],[\"1\",\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0021","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (8)x+(1)y=-2 與 (1)x+(-2)y=3 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"-17","explanation_zh":"D=-17。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046021,"parameters":{"rows":[[8,1,-2],[1,-2,3]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"8\",\"1\"],[\"1\",\"-2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0022","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (9)x+(1)y=-1 與 (1)x+(-1)y=-3 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"-10","explanation_zh":"D=-10。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046022,"parameters":{"rows":[[9,1,-1],[1,-1,-3]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"9\",\"1\"],[\"1\",\"-1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0023","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (10)x+(1)y=0 與 (1)x+(0)y=-2 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"D=-1。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046023,"parameters":{"rows":[[10,1,0],[1,0,-2]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"10\",\"1\"],[\"1\",\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0024","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (11)x+(1)y=1 與 (1)x+(1)y=-1 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"10","explanation_zh":"D=10。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046024,"parameters":{"rows":[[11,1,1],[1,1,-1]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"11\",\"1\"],[\"1\",\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0025","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的行列式","stem_zh":"實數論域中兩個限制 (12)x+(1)y=2 與 (1)x+(2)y=0 必須同時成立。係數矩陣的行列式 D=ae−bd 是多少？","answer":{"kind":"numeric","value":"23","explanation_zh":"D=23。主對角係數乘積減另一對角係數乘積；右側常數不參與行列式。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的行列式"]},"variant":{"family_id":"qf-linear-system-determinant","seed":202610046025,"parameters":{"rows":[[12,1,2],[1,2,0]],"domain":"reals","operation":"determinant","semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"12\",\"1\"],[\"1\",\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0026","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (0)x+(0)y=0 與 (0)x+(0)y=0 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無限多解","explanation_zh":"D=0；解分類=無限多解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046026,"parameters":{"rows":[[0,0,0],[0,0,0]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"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"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0027","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (0)x+(0)y=1 與 (0)x+(0)y=0 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無解","explanation_zh":"D=0；解分類=無解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046027,"parameters":{"rows":[[0,0,1],[0,0,0]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"0\",\"0\",\"0\"],[\"0\",\"0\",\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0028","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(0)y=2 與 (0)x+(0)y=0 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無限多解","explanation_zh":"D=0；解分類=無限多解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046028,"parameters":{"rows":[[1,0,2],[0,0,0]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"0\",\"0\",\"0\"],[\"1\",\"0\",\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0029","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(2)y=3 與 (2)x+(4)y=6 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無限多解","explanation_zh":"D=0；解分類=無限多解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046029,"parameters":{"rows":[[1,2,3],[2,4,6]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"2\",\"3\"],[\"1\",\"2\",\"3\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0030","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(2)y=3 與 (2)x+(4)y=7 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無解","explanation_zh":"D=0；解分類=無解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046030,"parameters":{"rows":[[1,2,3],[2,4,7]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"2\",\"3\"],[\"2\",\"4\",\"7\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0031","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (0)x+(1)y=2 與 (0)x+(2)y=5 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無解","explanation_zh":"D=0；解分類=無解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046031,"parameters":{"rows":[[0,1,2],[0,2,5]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"0\",\"1\",\"2\"],[\"0\",\"2\",\"5\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0032","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(0)y=0 與 (0)x+(-1)y=0 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"唯一解","explanation_zh":"D=-1；解分類=唯一解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046032,"parameters":{"rows":[[1,0,0],[0,-1,0]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"0\",\"1\",\"0\"],[\"1\",\"0\",\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0033","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(2)y=0 與 (0)x+(0)y=4 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無解","explanation_zh":"D=0；解分類=無解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046033,"parameters":{"rows":[[1,2,0],[0,0,4]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"0\",\"0\",\"1\"],[\"1\",\"2\",\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0034","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (-2)x+(1)y=1 與 (1)x+(-1)y=-2 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"唯一解","explanation_zh":"D=1；解分類=唯一解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046034,"parameters":{"rows":[[-2,1,1],[1,-1,-2]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"-1\",\"-2\"],[\"2\",\"-1\",\"-1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0035","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(3)y=5 與 (2)x+(6)y=10 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無限多解","explanation_zh":"D=0；解分類=無限多解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046035,"parameters":{"rows":[[1,3,5],[2,6,10]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"3\",\"5\"],[\"1\",\"3\",\"5\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0036","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(4)y=6 與 (2)x+(8)y=13 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無解","explanation_zh":"D=0；解分類=無解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046036,"parameters":{"rows":[[1,4,6],[2,8,13]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"4\",\"6\"],[\"2\",\"8\",\"13\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0037","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(1)y=-1 與 (1)x+(-1)y=1 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"唯一解","explanation_zh":"D=-2；解分類=唯一解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046037,"parameters":{"rows":[[1,1,-1],[1,-1,1]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"-1\",\"1\"],[\"1\",\"1\",\"-1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0038","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(6)y=8 與 (2)x+(12)y=16 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無限多解","explanation_zh":"D=0；解分類=無限多解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046038,"parameters":{"rows":[[1,6,8],[2,12,16]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"6\",\"8\"],[\"1\",\"6\",\"8\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0039","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(7)y=9 與 (2)x+(14)y=19 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無解","explanation_zh":"D=0；解分類=無解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046039,"parameters":{"rows":[[1,7,9],[2,14,19]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"7\",\"9\"],[\"2\",\"14\",\"19\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0040","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (4)x+(1)y=2 與 (1)x+(-1)y=-3 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"唯一解","explanation_zh":"D=-5；解分類=唯一解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046040,"parameters":{"rows":[[4,1,2],[1,-1,-3]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"-1\",\"-3\"],[\"4\",\"1\",\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0041","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(9)y=11 與 (2)x+(18)y=22 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無限多解","explanation_zh":"D=0；解分類=無限多解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046041,"parameters":{"rows":[[1,9,11],[2,18,22]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"9\",\"11\"],[\"1\",\"9\",\"11\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0042","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(10)y=12 與 (2)x+(20)y=25 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無解","explanation_zh":"D=0；解分類=無解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046042,"parameters":{"rows":[[1,10,12],[2,20,25]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"10\",\"12\"],[\"2\",\"20\",\"25\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0043","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (7)x+(1)y=0 與 (1)x+(-1)y=0 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"唯一解","explanation_zh":"D=-8；解分類=唯一解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046043,"parameters":{"rows":[[7,1,0],[1,-1,0]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"-1\",\"0\"],[\"7\",\"1\",\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0044","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(12)y=14 與 (2)x+(24)y=28 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無限多解","explanation_zh":"D=0；解分類=無限多解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046044,"parameters":{"rows":[[1,12,14],[2,24,28]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"12\",\"14\"],[\"1\",\"12\",\"14\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0045","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(13)y=15 與 (2)x+(26)y=31 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無解","explanation_zh":"D=0；解分類=無解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046045,"parameters":{"rows":[[1,13,15],[2,26,31]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"13\",\"15\"],[\"2\",\"26\",\"31\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0046","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (10)x+(1)y=-2 與 (1)x+(-1)y=3 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"唯一解","explanation_zh":"D=-11；解分類=唯一解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046046,"parameters":{"rows":[[10,1,-2],[1,-1,3]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"-1\",\"3\"],[\"10\",\"1\",\"-2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0047","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(15)y=17 與 (2)x+(30)y=34 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無限多解","explanation_zh":"D=0；解分類=無限多解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046047,"parameters":{"rows":[[1,15,17],[2,30,34]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"15\",\"17\"],[\"1\",\"15\",\"17\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0048","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(16)y=18 與 (2)x+(32)y=37 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無解","explanation_zh":"D=0；解分類=無解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046048,"parameters":{"rows":[[1,16,18],[2,32,37]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"16\",\"18\"],[\"2\",\"32\",\"37\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0049","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (13)x+(1)y=1 與 (1)x+(-1)y=-1 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"唯一解","explanation_zh":"D=-14；解分類=唯一解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046049,"parameters":{"rows":[[13,1,1],[1,-1,-1]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"-1\",\"-1\"],[\"13\",\"1\",\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0050","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"二元線性系統的解集合分類","stem_zh":"實數論域中兩個限制 (1)x+(18)y=20 與 (2)x+(36)y=40 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。","answer":{"kind":"short_text","value":"無限多解","explanation_zh":"D=0；解分類=無限多解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二元線性系統的解集合分類"]},"variant":{"family_id":"qf-linear-system-solution-class","seed":202610046050,"parameters":{"rows":[[1,18,20],[2,36,40]],"domain":"reals","operation":"solution_class","semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"18\",\"20\"],[\"1\",\"18\",\"20\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0051","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (0)x+(0)y=0 與 (0)x+(0)y=0 必須同時成立。最簡正分母候選 x=0/1、y=0/1 是否滿足整個系統？","answer":{"kind":"short_text","value":"是","explanation_zh":"分母清除殘差=(0, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046051,"parameters":{"rows":[[0,0,0],[0,0,0]],"domain":"reals","operation":"verify_solution","x":[0,1],"y":[0,1],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"0\",\"0\",\"0\"],[\"0\",\"0\",\"0\"]],\"x\":[0,1],\"y\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0052","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (0)x+(0)y=1 與 (0)x+(0)y=0 必須同時成立。最簡正分母候選 x=0/1、y=0/1 是否滿足整個系統？","answer":{"kind":"short_text","value":"否","explanation_zh":"分母清除殘差=(-1, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046052,"parameters":{"rows":[[0,0,1],[0,0,0]],"domain":"reals","operation":"verify_solution","x":[0,1],"y":[0,1],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"0\",\"0\",\"0\"],[\"0\",\"0\",\"1\"]],\"x\":[0,1],\"y\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0053","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (1)x+(2)y=0 與 (2)x+(4)y=0 必須同時成立。最簡正分母候選 x=2/1、y=-1/1 是否滿足整個系統？","answer":{"kind":"short_text","value":"是","explanation_zh":"分母清除殘差=(0, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046053,"parameters":{"rows":[[1,2,0],[2,4,0]],"domain":"reals","operation":"verify_solution","x":[2,1],"y":[-1,1],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"1\",\"2\",\"0\"],[\"1\",\"2\",\"0\"]],\"x\":[2,1],\"y\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0054","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (1)x+(0)y=0 與 (0)x+(1)y=1 必須同時成立。最簡正分母候選 x=0/1、y=0/1 是否滿足整個系統？","answer":{"kind":"short_text","value":"否","explanation_zh":"分母清除殘差=(0, -1)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046054,"parameters":{"rows":[[1,0,0],[0,1,1]],"domain":"reals","operation":"verify_solution","x":[0,1],"y":[0,1],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"0\",\"1\",\"1\"],[\"1\",\"0\",\"0\"]],\"x\":[0,1],\"y\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0055","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=-7 與 (4)x+(-3)y=-17 必須同時成立。最簡正分母候選 x=-4/1、y=1/3 是否滿足整個系統？","answer":{"kind":"short_text","value":"是","explanation_zh":"分母清除殘差=(0, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046055,"parameters":{"rows":[[2,3,-7],[4,-3,-17]],"domain":"reals","operation":"verify_solution","x":[-4,1],"y":[1,3],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-7\"],[\"4\",\"-3\",\"-17\"]],\"x\":[-4,1],\"y\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0056","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=-5 與 (4)x+(-3)y=-16 必須同時成立。最簡正分母候選 x=-3/1、y=2/3 是否滿足整個系統？","answer":{"kind":"short_text","value":"否","explanation_zh":"分母清除殘差=(3, 6)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046056,"parameters":{"rows":[[2,3,-5],[4,-3,-16]],"domain":"reals","operation":"verify_solution","x":[-3,1],"y":[2,3],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-5\"],[\"4\",\"-3\",\"-16\"]],\"x\":[-3,1],\"y\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0057","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=-3 與 (4)x+(-3)y=-15 必須同時成立。最簡正分母候選 x=-3/1、y=1/1 是否滿足整個系統？","answer":{"kind":"short_text","value":"是","explanation_zh":"分母清除殘差=(0, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046057,"parameters":{"rows":[[2,3,-3],[4,-3,-15]],"domain":"reals","operation":"verify_solution","x":[-3,1],"y":[1,1],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-3\"],[\"4\",\"-3\",\"-15\"]],\"x\":[-3,1],\"y\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0058","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=-8 與 (4)x+(-3)y=-7 必須同時成立。最簡正分母候選 x=-2/1、y=-1/1 是否滿足整個系統？","answer":{"kind":"short_text","value":"否","explanation_zh":"分母清除殘差=(1, 2)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046058,"parameters":{"rows":[[2,3,-8],[4,-3,-7]],"domain":"reals","operation":"verify_solution","x":[-2,1],"y":[-1,1],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-8\"],[\"4\",\"-3\",\"-7\"]],\"x\":[-2,1],\"y\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0059","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=-6 與 (4)x+(-3)y=-6 必須同時成立。最簡正分母候選 x=-2/1、y=-2/3 是否滿足整個系統？","answer":{"kind":"short_text","value":"是","explanation_zh":"分母清除殘差=(0, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046059,"parameters":{"rows":[[2,3,-6],[4,-3,-6]],"domain":"reals","operation":"verify_solution","x":[-2,1],"y":[-2,3],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-6\"],[\"4\",\"-3\",\"-6\"]],\"x\":[-2,1],\"y\":[-2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0060","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=-4 與 (4)x+(-3)y=-5 必須同時成立。最簡正分母候選 x=-1/1、y=-1/3 是否滿足整個系統？","answer":{"kind":"short_text","value":"否","explanation_zh":"分母清除殘差=(3, 6)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046060,"parameters":{"rows":[[2,3,-4],[4,-3,-5]],"domain":"reals","operation":"verify_solution","x":[-1,1],"y":[-1,3],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-4\"],[\"4\",\"-3\",\"-5\"]],\"x\":[-1,1],\"y\":[-1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0061","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=-2 與 (4)x+(-3)y=-4 必須同時成立。最簡正分母候選 x=-1/1、y=0/1 是否滿足整個系統？","answer":{"kind":"short_text","value":"是","explanation_zh":"分母清除殘差=(0, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046061,"parameters":{"rows":[[2,3,-2],[4,-3,-4]],"domain":"reals","operation":"verify_solution","x":[-1,1],"y":[0,1],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-2\"],[\"4\",\"-3\",\"-4\"]],\"x\":[-1,1],\"y\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0062","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=0 與 (4)x+(-3)y=-3 必須同時成立。最簡正分母候選 x=0/1、y=1/3 是否滿足整個系統？","answer":{"kind":"short_text","value":"否","explanation_zh":"分母清除殘差=(3, 6)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046062,"parameters":{"rows":[[2,3,0],[4,-3,-3]],"domain":"reals","operation":"verify_solution","x":[0,1],"y":[1,3],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"0\"],[\"4\",\"-3\",\"-3\"]],\"x\":[0,1],\"y\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0063","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=2 與 (4)x+(-3)y=-2 必須同時成立。最簡正分母候選 x=0/1、y=2/3 是否滿足整個系統？","answer":{"kind":"short_text","value":"是","explanation_zh":"分母清除殘差=(0, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046063,"parameters":{"rows":[[2,3,2],[4,-3,-2]],"domain":"reals","operation":"verify_solution","x":[0,1],"y":[2,3],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"2\"],[\"4\",\"-3\",\"-2\"]],\"x\":[0,1],\"y\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0064","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=4 與 (4)x+(-3)y=-1 必須同時成立。最簡正分母候選 x=1/1、y=1/1 是否滿足整個系統？","answer":{"kind":"short_text","value":"否","explanation_zh":"分母清除殘差=(1, 2)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046064,"parameters":{"rows":[[2,3,4],[4,-3,-1]],"domain":"reals","operation":"verify_solution","x":[1,1],"y":[1,1],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"4\"],[\"4\",\"-3\",\"-1\"]],\"x\":[1,1],\"y\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0065","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=-1 與 (4)x+(-3)y=7 必須同時成立。最簡正分母候選 x=1/1、y=-1/1 是否滿足整個系統？","answer":{"kind":"short_text","value":"是","explanation_zh":"分母清除殘差=(0, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046065,"parameters":{"rows":[[2,3,-1],[4,-3,7]],"domain":"reals","operation":"verify_solution","x":[1,1],"y":[-1,1],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-1\"],[\"4\",\"-3\",\"7\"]],\"x\":[1,1],\"y\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0066","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=1 與 (4)x+(-3)y=8 必須同時成立。最簡正分母候選 x=2/1、y=-2/3 是否滿足整個系統？","answer":{"kind":"short_text","value":"否","explanation_zh":"分母清除殘差=(3, 6)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046066,"parameters":{"rows":[[2,3,1],[4,-3,8]],"domain":"reals","operation":"verify_solution","x":[2,1],"y":[-2,3],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"1\"],[\"4\",\"-3\",\"8\"]],\"x\":[2,1],\"y\":[-2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0067","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=3 與 (4)x+(-3)y=9 必須同時成立。最簡正分母候選 x=2/1、y=-1/3 是否滿足整個系統？","answer":{"kind":"short_text","value":"是","explanation_zh":"分母清除殘差=(0, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046067,"parameters":{"rows":[[2,3,3],[4,-3,9]],"domain":"reals","operation":"verify_solution","x":[2,1],"y":[-1,3],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"3\"],[\"4\",\"-3\",\"9\"]],\"x\":[2,1],\"y\":[-1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0068","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=5 與 (4)x+(-3)y=10 必須同時成立。最簡正分母候選 x=3/1、y=0/1 是否滿足整個系統？","answer":{"kind":"short_text","value":"否","explanation_zh":"分母清除殘差=(1, 2)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046068,"parameters":{"rows":[[2,3,5],[4,-3,10]],"domain":"reals","operation":"verify_solution","x":[3,1],"y":[0,1],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"5\"],[\"4\",\"-3\",\"10\"]],\"x\":[3,1],\"y\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0069","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=7 與 (4)x+(-3)y=11 必須同時成立。最簡正分母候選 x=3/1、y=1/3 是否滿足整個系統？","answer":{"kind":"short_text","value":"是","explanation_zh":"分母清除殘差=(0, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046069,"parameters":{"rows":[[2,3,7],[4,-3,11]],"domain":"reals","operation":"verify_solution","x":[3,1],"y":[1,3],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"7\"],[\"4\",\"-3\",\"11\"]],\"x\":[3,1],\"y\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0070","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=9 與 (4)x+(-3)y=12 必須同時成立。最簡正分母候選 x=4/1、y=2/3 是否滿足整個系統？","answer":{"kind":"short_text","value":"否","explanation_zh":"分母清除殘差=(3, 6)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046070,"parameters":{"rows":[[2,3,9],[4,-3,12]],"domain":"reals","operation":"verify_solution","x":[4,1],"y":[2,3],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"9\"],[\"4\",\"-3\",\"12\"]],\"x\":[4,1],\"y\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0071","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=11 與 (4)x+(-3)y=13 必須同時成立。最簡正分母候選 x=4/1、y=1/1 是否滿足整個系統？","answer":{"kind":"short_text","value":"是","explanation_zh":"分母清除殘差=(0, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046071,"parameters":{"rows":[[2,3,11],[4,-3,13]],"domain":"reals","operation":"verify_solution","x":[4,1],"y":[1,1],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"11\"],[\"4\",\"-3\",\"13\"]],\"x\":[4,1],\"y\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0072","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=6 與 (4)x+(-3)y=21 必須同時成立。最簡正分母候選 x=5/1、y=-1/1 是否滿足整個系統？","answer":{"kind":"short_text","value":"否","explanation_zh":"分母清除殘差=(1, 2)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046072,"parameters":{"rows":[[2,3,6],[4,-3,21]],"domain":"reals","operation":"verify_solution","x":[5,1],"y":[-1,1],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"6\"],[\"4\",\"-3\",\"21\"]],\"x\":[5,1],\"y\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0073","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=8 與 (4)x+(-3)y=22 必須同時成立。最簡正分母候選 x=5/1、y=-2/3 是否滿足整個系統？","answer":{"kind":"short_text","value":"是","explanation_zh":"分母清除殘差=(0, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046073,"parameters":{"rows":[[2,3,8],[4,-3,22]],"domain":"reals","operation":"verify_solution","x":[5,1],"y":[-2,3],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"8\"],[\"4\",\"-3\",\"22\"]],\"x\":[5,1],\"y\":[-2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0074","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=10 與 (4)x+(-3)y=23 必須同時成立。最簡正分母候選 x=6/1、y=-1/3 是否滿足整個系統？","answer":{"kind":"short_text","value":"否","explanation_zh":"分母清除殘差=(3, 6)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046074,"parameters":{"rows":[[2,3,10],[4,-3,23]],"domain":"reals","operation":"verify_solution","x":[6,1],"y":[-1,3],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"10\"],[\"4\",\"-3\",\"23\"]],\"x\":[6,1],\"y\":[-1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0075","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"精確有理候選的雙列驗證","stem_zh":"實數論域中兩個限制 (2)x+(3)y=12 與 (4)x+(-3)y=24 必須同時成立。最簡正分母候選 x=6/1、y=0/1 是否滿足整個系統？","answer":{"kind":"short_text","value":"是","explanation_zh":"分母清除殘差=(0, 0)。兩列殘差都為零才接受；只通過其中一列仍不構成系統解。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["精確有理候選的雙列驗證"]},"variant":{"family_id":"qf-linear-system-solution-verification","seed":202610046075,"parameters":{"rows":[[2,3,12],[4,-3,24]],"domain":"reals","operation":"verify_solution","x":[6,1],"y":[0,1],"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"12\"],[\"4\",\"-3\",\"24\"]],\"x\":[6,1],\"y\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0076","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(0)y=0 與 (0)x+(1)y=0 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(0/1, 0/1)","explanation_zh":"D=1；解=(0/1, 0/1)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046076,"parameters":{"rows":[[1,0,0],[0,1,0]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"0\",\"1\",\"0\"],[\"1\",\"0\",\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0077","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=1 與 (1)x+(-1)y=0 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(1/2, 1/2)","explanation_zh":"D=-2；解=(1/2, 1/2)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046077,"parameters":{"rows":[[1,1,1],[1,-1,0]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-1\",\"0\"],[\"1\",\"1\",\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0078","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (-2)x+(1)y=3 與 (1)x+(2)y=-4 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(-2/1, -1/1)","explanation_zh":"D=-5；解=(-2/1, -1/1)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046078,"parameters":{"rows":[[-2,1,3],[1,2,-4]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"2\",\"-4\"],[\"2\",\"-1\",\"-3\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0079","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=-9 與 (1)x+(-2)y=1 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(-17/3, -10/3)","explanation_zh":"D=-3；解=(-17/3, -10/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046079,"parameters":{"rows":[[1,1,-9],[1,-2,1]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"1\"],[\"1\",\"1\",\"-9\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0080","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=-8 與 (1)x+(-2)y=2 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(-14/3, -10/3)","explanation_zh":"D=-3；解=(-14/3, -10/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046080,"parameters":{"rows":[[1,1,-8],[1,-2,2]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"2\"],[\"1\",\"1\",\"-8\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0081","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=-7 與 (1)x+(-2)y=-2 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(-16/3, -5/3)","explanation_zh":"D=-3；解=(-16/3, -5/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046081,"parameters":{"rows":[[1,1,-7],[1,-2,-2]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-2\"],[\"1\",\"1\",\"-7\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0082","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=-6 與 (1)x+(-2)y=-1 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(-13/3, -5/3)","explanation_zh":"D=-3；解=(-13/3, -5/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046082,"parameters":{"rows":[[1,1,-6],[1,-2,-1]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-1\"],[\"1\",\"1\",\"-6\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0083","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=-5 與 (1)x+(-2)y=0 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(-10/3, -5/3)","explanation_zh":"D=-3；解=(-10/3, -5/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046083,"parameters":{"rows":[[1,1,-5],[1,-2,0]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"0\"],[\"1\",\"1\",\"-5\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0084","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=-4 與 (1)x+(-2)y=1 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(-7/3, -5/3)","explanation_zh":"D=-3；解=(-7/3, -5/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046084,"parameters":{"rows":[[1,1,-4],[1,-2,1]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"1\"],[\"1\",\"1\",\"-4\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0085","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=-3 與 (1)x+(-2)y=2 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(-4/3, -5/3)","explanation_zh":"D=-3；解=(-4/3, -5/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046085,"parameters":{"rows":[[1,1,-3],[1,-2,2]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"2\"],[\"1\",\"1\",\"-3\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0086","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=-2 與 (1)x+(-2)y=-2 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(-2/1, 0/1)","explanation_zh":"D=-3；解=(-2/1, 0/1)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046086,"parameters":{"rows":[[1,1,-2],[1,-2,-2]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-2\"],[\"1\",\"1\",\"-2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0087","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=-1 與 (1)x+(-2)y=-1 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(-1/1, 0/1)","explanation_zh":"D=-3；解=(-1/1, 0/1)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046087,"parameters":{"rows":[[1,1,-1],[1,-2,-1]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-1\"],[\"1\",\"1\",\"-1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0088","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=0 與 (1)x+(-2)y=0 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(0/1, 0/1)","explanation_zh":"D=-3；解=(0/1, 0/1)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046088,"parameters":{"rows":[[1,1,0],[1,-2,0]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"0\"],[\"1\",\"1\",\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0089","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=1 與 (1)x+(-2)y=1 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(1/1, 0/1)","explanation_zh":"D=-3；解=(1/1, 0/1)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046089,"parameters":{"rows":[[1,1,1],[1,-2,1]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"1\"],[\"1\",\"1\",\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0090","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=2 與 (1)x+(-2)y=2 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(2/1, 0/1)","explanation_zh":"D=-3；解=(2/1, 0/1)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046090,"parameters":{"rows":[[1,1,2],[1,-2,2]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"2\"],[\"1\",\"1\",\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0091","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=3 與 (1)x+(-2)y=-2 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(4/3, 5/3)","explanation_zh":"D=-3；解=(4/3, 5/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046091,"parameters":{"rows":[[1,1,3],[1,-2,-2]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-2\"],[\"1\",\"1\",\"3\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0092","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=4 與 (1)x+(-2)y=-1 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(7/3, 5/3)","explanation_zh":"D=-3；解=(7/3, 5/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046092,"parameters":{"rows":[[1,1,4],[1,-2,-1]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-1\"],[\"1\",\"1\",\"4\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0093","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=5 與 (1)x+(-2)y=0 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(10/3, 5/3)","explanation_zh":"D=-3；解=(10/3, 5/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046093,"parameters":{"rows":[[1,1,5],[1,-2,0]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"0\"],[\"1\",\"1\",\"5\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0094","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=6 與 (1)x+(-2)y=1 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(13/3, 5/3)","explanation_zh":"D=-3；解=(13/3, 5/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046094,"parameters":{"rows":[[1,1,6],[1,-2,1]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"1\"],[\"1\",\"1\",\"6\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0095","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=7 與 (1)x+(-2)y=2 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(16/3, 5/3)","explanation_zh":"D=-3；解=(16/3, 5/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046095,"parameters":{"rows":[[1,1,7],[1,-2,2]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"2\"],[\"1\",\"1\",\"7\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0096","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=8 與 (1)x+(-2)y=-2 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(14/3, 10/3)","explanation_zh":"D=-3；解=(14/3, 10/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046096,"parameters":{"rows":[[1,1,8],[1,-2,-2]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-2\"],[\"1\",\"1\",\"8\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0097","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=9 與 (1)x+(-2)y=-1 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(17/3, 10/3)","explanation_zh":"D=-3；解=(17/3, 10/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046097,"parameters":{"rows":[[1,1,9],[1,-2,-1]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-1\"],[\"1\",\"1\",\"9\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0098","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=10 與 (1)x+(-2)y=0 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(20/3, 10/3)","explanation_zh":"D=-3；解=(20/3, 10/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046098,"parameters":{"rows":[[1,1,10],[1,-2,0]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"0\"],[\"1\",\"1\",\"10\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0099","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=11 與 (1)x+(-2)y=1 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(23/3, 10/3)","explanation_zh":"D=-3；解=(23/3, 10/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046099,"parameters":{"rows":[[1,1,11],[1,-2,1]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"1\"],[\"1\",\"1\",\"11\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r06-system-linear-equations-0100","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","language":"zh-Hant","source_mko_ids":["mko-system-linear-equations"],"title_zh":"非零行列式的精確唯一解","stem_zh":"實數論域中兩個限制 (1)x+(1)y=12 與 (1)x+(-2)y=2 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。","answer":{"kind":"expression","value":"(26/3, 10/3)","explanation_zh":"D=-3；解=(26/3, 10/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非零行列式的精確唯一解"]},"variant":{"family_id":"qf-linear-system-unique-solve","seed":202610046100,"parameters":{"rows":[[1,1,12],[1,-2,2]],"domain":"reals","operation":"unique_solve","semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"2\"],[\"1\",\"1\",\"12\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}