{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0001","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[0]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([1, 1], [[0]], [[0]], 是)","explanation_zh":"有限結構核對=([1, 1], [[0]], [[0]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162001,"parameters":{"M":[[0]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0002","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-11, -12]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([1, 2], [[-11], [-12]], [[-11, -12]], 是)","explanation_zh":"有限結構核對=([1, 2], [[-11], [-12]], [[-11, -12]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162002,"parameters":{"M":[[-11,-12]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-11,-12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0003","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-10, -11, -12]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([1, 3], [[-10], [-11], [-12]], [[-10, -11, -12]], 是)","explanation_zh":"有限結構核對=([1, 3], [[-10], [-11], [-12]], [[-10, -11, -12]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162003,"parameters":{"M":[[-10,-11,-12]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-10,-11,-12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0004","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-9, -10, -11, -12]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([1, 4], [[-9], [-10], [-11], [-12]], [[-9, -10, -11, -12]], 是)","explanation_zh":"有限結構核對=([1, 4], [[-9], [-10], [-11], [-12]], [[-9, -10, -11, -12]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162004,"parameters":{"M":[[-9,-10,-11,-12]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-9,-10,-11,-12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0005","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-8, -9, -10, -11, -12]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([1, 5], [[-8], [-9], [-10], [-11], [-12]], [[-8, -9, -10, -11, -12]], 是)","explanation_zh":"有限結構核對=([1, 5], [[-8], [-9], [-10], [-11], [-12]], [[-8, -9, -10, -11, -12]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162005,"parameters":{"M":[[-8,-9,-10,-11,-12]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-8,-9,-10,-11,-12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0006","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-7], [-6]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([2, 1], [[-7, -6]], [[-7], [-6]], 是)","explanation_zh":"有限結構核對=([2, 1], [[-7, -6]], [[-7], [-6]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162006,"parameters":{"M":[[-7],[-6]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-7],[-6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0007","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-6, -7], [-5, -6]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([2, 2], [[-6, -5], [-7, -6]], [[-6, -7], [-5, -6]], 是)","explanation_zh":"有限結構核對=([2, 2], [[-6, -5], [-7, -6]], [[-6, -7], [-5, -6]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162007,"parameters":{"M":[[-6,-7],[-5,-6]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-6,-7],[-5,-6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0008","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-5, -6, -7], [-4, -5, -6]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([2, 3], [[-5, -4], [-6, -5], [-7, -6]], [[-5, -6, -7], [-4, -5, -6]], 是)","explanation_zh":"有限結構核對=([2, 3], [[-5, -4], [-6, -5], [-7, -6]], [[-5, -6, -7], [-4, -5, -6]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162008,"parameters":{"M":[[-5,-6,-7],[-4,-5,-6]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-5,-6,-7],[-4,-5,-6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0009","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-4, -5, -6, -7], [-3, -4, -5, -6]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([2, 4], [[-4, -3], [-5, -4], [-6, -5], [-7, -6]], [[-4, -5, -6, -7], [-3, -4, -5, -6]], 是)","explanation_zh":"有限結構核對=([2, 4], [[-4, -3], [-5, -4], [-6, -5], [-7, -6]], [[-4, -5, -6, -7], [-3, -4, -5, -6]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162009,"parameters":{"M":[[-4,-5,-6,-7],[-3,-4,-5,-6]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-4,-5,-6,-7],[-3,-4,-5,-6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0010","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([2, 5], [[-3, -2], [-4, -3], [-5, -4], [-6, -5], [-7, -6]], [[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]], 是)","explanation_zh":"有限結構核對=([2, 5], [[-3, -2], [-4, -3], [-5, -4], [-6, -5], [-7, -6]], [[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162010,"parameters":{"M":[[-3,-4,-5,-6,-7],[-2,-3,-4,-5,-6]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-3,-4,-5,-6,-7],[-2,-3,-4,-5,-6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0011","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-2], [-1], [0]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([3, 1], [[-2, -1, 0]], [[-2], [-1], [0]], 是)","explanation_zh":"有限結構核對=([3, 1], [[-2, -1, 0]], [[-2], [-1], [0]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162011,"parameters":{"M":[[-2],[-1],[0]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-2],[-1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0012","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-1, -2], [0, -1], [1, 0]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([3, 2], [[-1, 0, 1], [-2, -1, 0]], [[-1, -2], [0, -1], [1, 0]], 是)","explanation_zh":"有限結構核對=([3, 2], [[-1, 0, 1], [-2, -1, 0]], [[-1, -2], [0, -1], [1, 0]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162012,"parameters":{"M":[[-1,-2],[0,-1],[1,0]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-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","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0013","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[0, -1, -2], [1, 0, -1], [2, 1, 0]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([3, 3], [[0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[0, -1, -2], [1, 0, -1], [2, 1, 0]], 是)","explanation_zh":"有限結構核對=([3, 3], [[0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[0, -1, -2], [1, 0, -1], [2, 1, 0]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162013,"parameters":{"M":[[0,-1,-2],[1,0,-1],[2,1,0]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[0,-1,-2],[1,0,-1],[2,1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0014","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([3, 4], [[1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]], 是)","explanation_zh":"有限結構核對=([3, 4], [[1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162014,"parameters":{"M":[[1,0,-1,-2],[2,1,0,-1],[3,2,1,0]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[1,0,-1,-2],[2,1,0,-1],[3,2,1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0015","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([3, 5], [[2, 3, 4], [1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]], 是)","explanation_zh":"有限結構核對=([3, 5], [[2, 3, 4], [1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162015,"parameters":{"M":[[2,1,0,-1,-2],[3,2,1,0,-1],[4,3,2,1,0]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[2,1,0,-1,-2],[3,2,1,0,-1],[4,3,2,1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0016","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[3], [4], [5], [6]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([4, 1], [[3, 4, 5, 6]], [[3], [4], [5], [6]], 是)","explanation_zh":"有限結構核對=([4, 1], [[3, 4, 5, 6]], [[3], [4], [5], [6]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162016,"parameters":{"M":[[3],[4],[5],[6]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[3],[4],[5],[6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0017","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[4, 3], [5, 4], [6, 5], [7, 6]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([4, 2], [[4, 5, 6, 7], [3, 4, 5, 6]], [[4, 3], [5, 4], [6, 5], [7, 6]], 是)","explanation_zh":"有限結構核對=([4, 2], [[4, 5, 6, 7], [3, 4, 5, 6]], [[4, 3], [5, 4], [6, 5], [7, 6]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162017,"parameters":{"M":[[4,3],[5,4],[6,5],[7,6]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[4,3],[5,4],[6,5],[7,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0018","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([4, 3], [[5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]], 是)","explanation_zh":"有限結構核對=([4, 3], [[5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162018,"parameters":{"M":[[5,4,3],[6,5,4],[7,6,5],[8,7,6]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[5,4,3],[6,5,4],[7,6,5],[8,7,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0019","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([4, 4], [[6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]], 是)","explanation_zh":"有限結構核對=([4, 4], [[6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162019,"parameters":{"M":[[6,5,4,3],[7,6,5,4],[8,7,6,5],[9,8,7,6]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[6,5,4,3],[7,6,5,4],[8,7,6,5],[9,8,7,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0020","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([4, 5], [[7, 8, 9, 10], [6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]], 是)","explanation_zh":"有限結構核對=([4, 5], [[7, 8, 9, 10], [6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162020,"parameters":{"M":[[7,6,5,4,3],[8,7,6,5,4],[9,8,7,6,5],[10,9,8,7,6]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[7,6,5,4,3],[8,7,6,5,4],[9,8,7,6,5],[10,9,8,7,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0021","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[8], [9], [10], [11], [12]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([5, 1], [[8, 9, 10, 11, 12]], [[8], [9], [10], [11], [12]], 是)","explanation_zh":"有限結構核對=([5, 1], [[8, 9, 10, 11, 12]], [[8], [9], [10], [11], [12]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162021,"parameters":{"M":[[8],[9],[10],[11],[12]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[8],[9],[10],[11],[12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0022","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([5, 2], [[9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]], 是)","explanation_zh":"有限結構核對=([5, 2], [[9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162022,"parameters":{"M":[[9,8],[10,9],[11,10],[12,11],[13,12]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[9,8],[10,9],[11,10],[12,11],[13,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0023","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([5, 3], [[10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]], 是)","explanation_zh":"有限結構核對=([5, 3], [[10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162023,"parameters":{"M":[[10,9,8],[11,10,9],[12,11,10],[13,12,11],[14,13,12]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[10,9,8],[11,10,9],[12,11,10],[13,12,11],[14,13,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0024","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([5, 4], [[11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]], 是)","explanation_zh":"有限結構核對=([5, 4], [[11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162024,"parameters":{"M":[[11,10,9,8],[12,11,10,9],[13,12,11,10],[14,13,12,11],[15,14,13,12]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[11,10,9,8],[12,11,10,9],[13,12,11,10],[14,13,12,11],[15,14,13,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0025","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"矩陣轉置的往返","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]]。求 (M尺寸, Mᵀ, (Mᵀ)ᵀ, 往返與M相同資格)。","answer":{"kind":"expression","value":"([5, 5], [[12, 13, 14, 15, 16], [11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]], 是)","explanation_zh":"有限結構核對=([5, 5], [[12, 13, 14, 15, 16], [11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]], 是)。交換行列位置兩次，逐格核往返，不反轉元素順序。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣轉置的往返"]},"variant":{"family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162025,"parameters":{"M":[[12,11,10,9,8],[13,12,11,10,9],[14,13,12,11,10],[15,14,13,12,11],[16,15,14,13,12]],"domain":"Z_rect_1based","operation":"matrix-transpose-roundtrip","semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[12,11,10,9,8],[13,12,11,10,9],[14,13,12,11,10],[15,14,13,12,11],[16,15,14,13,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0026","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[0]]、N=[[0]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([1, 1], [[0]], [[0]], [[0]])","explanation_zh":"有限結構核對=([1, 1], [[0]], [[0]], [[0]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162026,"parameters":{"M":[[0]],"N":[[0]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[0]],\"N\":[[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0027","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[-11, -12]]、N=[[0, 1]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([1, 2], [[-11, -11]], [[-11], [-11]], [[-11], [-11]])","explanation_zh":"有限結構核對=([1, 2], [[-11, -11]], [[-11], [-11]], [[-11], [-11]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162027,"parameters":{"M":[[-11,-12]],"N":[[0,1]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-11,-12]],\"N\":[[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0028","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[-10, -11, -12]]、N=[[0, 1, 2]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([1, 3], [[-10, -10, -10]], [[-10], [-10], [-10]], [[-10], [-10], [-10]])","explanation_zh":"有限結構核對=([1, 3], [[-10, -10, -10]], [[-10], [-10], [-10]], [[-10], [-10], [-10]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162028,"parameters":{"M":[[-10,-11,-12]],"N":[[0,1,2]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-10,-11,-12]],\"N\":[[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0029","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[-9, -10, -11, -12]]、N=[[0, 1, 2, 3]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([1, 4], [[-9, -9, -9, -9]], [[-9], [-9], [-9], [-9]], [[-9], [-9], [-9], [-9]])","explanation_zh":"有限結構核對=([1, 4], [[-9, -9, -9, -9]], [[-9], [-9], [-9], [-9]], [[-9], [-9], [-9], [-9]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162029,"parameters":{"M":[[-9,-10,-11,-12]],"N":[[0,1,2,3]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-9,-10,-11,-12]],\"N\":[[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0030","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[-8, -9, -10, -11, -12]]、N=[[0, 1, 2, 3, 4]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([1, 5], [[-8, -8, -8, -8, -8]], [[-8], [-8], [-8], [-8], [-8]], [[-8], [-8], [-8], [-8], [-8]])","explanation_zh":"有限結構核對=([1, 5], [[-8, -8, -8, -8, -8]], [[-8], [-8], [-8], [-8], [-8]], [[-8], [-8], [-8], [-8], [-8]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162030,"parameters":{"M":[[-8,-9,-10,-11,-12]],"N":[[0,1,2,3,4]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-8,-9,-10,-11,-12]],\"N\":[[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0031","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[-7], [-6]]、N=[[0], [-1]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([2, 1], [[-7], [-7]], [[-7, -7]], [[-7, -7]])","explanation_zh":"有限結構核對=([2, 1], [[-7], [-7]], [[-7, -7]], [[-7, -7]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162031,"parameters":{"M":[[-7],[-6]],"N":[[0],[-1]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-7],[-6]],\"N\":[[0],[-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0032","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[-6, -7], [-5, -6]]、N=[[0, 1], [-1, 0]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([2, 2], [[-6, -6], [-6, -6]], [[-6, -6], [-6, -6]], [[-6, -6], [-6, -6]])","explanation_zh":"有限結構核對=([2, 2], [[-6, -6], [-6, -6]], [[-6, -6], [-6, -6]], [[-6, -6], [-6, -6]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162032,"parameters":{"M":[[-6,-7],[-5,-6]],"N":[[0,1],[-1,0]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-6,-7],[-5,-6]],\"N\":[[0,1],[-1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0033","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[-5, -6, -7], [-4, -5, -6]]、N=[[0, 1, 2], [-1, 0, 1]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([2, 3], [[-5, -5, -5], [-5, -5, -5]], [[-5, -5], [-5, -5], [-5, -5]], [[-5, -5], [-5, -5], [-5, -5]])","explanation_zh":"有限結構核對=([2, 3], [[-5, -5, -5], [-5, -5, -5]], [[-5, -5], [-5, -5], [-5, -5]], [[-5, -5], [-5, -5], [-5, -5]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162033,"parameters":{"M":[[-5,-6,-7],[-4,-5,-6]],"N":[[0,1,2],[-1,0,1]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-5,-6,-7],[-4,-5,-6]],\"N\":[[0,1,2],[-1,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0034","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[-4, -5, -6, -7], [-3, -4, -5, -6]]、N=[[0, 1, 2, 3], [-1, 0, 1, 2]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([2, 4], [[-4, -4, -4, -4], [-4, -4, -4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]])","explanation_zh":"有限結構核對=([2, 4], [[-4, -4, -4, -4], [-4, -4, -4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162034,"parameters":{"M":[[-4,-5,-6,-7],[-3,-4,-5,-6]],"N":[[0,1,2,3],[-1,0,1,2]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-4,-5,-6,-7],[-3,-4,-5,-6]],\"N\":[[0,1,2,3],[-1,0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0035","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]]、N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([2, 5], [[-3, -3, -3, -3, -3], [-3, -3, -3, -3, -3]], [[-3, -3], [-3, -3], [-3, -3], [-3, -3], [-3, -3]], [[-3, -3], [-3, -3], [-3, -3], [-3, -3], [-3, -3]])","explanation_zh":"有限結構核對=([2, 5], [[-3, -3, -3, -3, -3], [-3, -3, -3, -3, -3]], [[-3, -3], [-3, -3], [-3, -3], [-3, -3], [-3, -3]], [[-3, -3], [-3, -3], [-3, -3], [-3, -3], [-3, -3]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162035,"parameters":{"M":[[-3,-4,-5,-6,-7],[-2,-3,-4,-5,-6]],"N":[[0,1,2,3,4],[-1,0,1,2,3]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-3,-4,-5,-6,-7],[-2,-3,-4,-5,-6]],\"N\":[[0,1,2,3,4],[-1,0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0036","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[-2], [-1], [0]]、N=[[0], [-1], [-2]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([3, 1], [[-2], [-2], [-2]], [[-2, -2, -2]], [[-2, -2, -2]])","explanation_zh":"有限結構核對=([3, 1], [[-2], [-2], [-2]], [[-2, -2, -2]], [[-2, -2, -2]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162036,"parameters":{"M":[[-2],[-1],[0]],"N":[[0],[-1],[-2]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-2],[-1],[0]],\"N\":[[0],[-1],[-2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0037","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[-1, -2], [0, -1], [1, 0]]、N=[[0, 1], [-1, 0], [-2, -1]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([3, 2], [[-1, -1], [-1, -1], [-1, -1]], [[-1, -1, -1], [-1, -1, -1]], [[-1, -1, -1], [-1, -1, -1]])","explanation_zh":"有限結構核對=([3, 2], [[-1, -1], [-1, -1], [-1, -1]], [[-1, -1, -1], [-1, -1, -1]], [[-1, -1, -1], [-1, -1, -1]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162037,"parameters":{"M":[[-1,-2],[0,-1],[1,0]],"N":[[0,1],[-1,0],[-2,-1]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-1,-2],[0,-1],[1,0]],\"N\":[[0,1],[-1,0],[-2,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0038","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[0, -1, -2], [1, 0, -1], [2, 1, 0]]、N=[[0, 1, 2], [-1, 0, 1], [-2, -1, 0]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([3, 3], [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 0]])","explanation_zh":"有限結構核對=([3, 3], [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 0]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162038,"parameters":{"M":[[0,-1,-2],[1,0,-1],[2,1,0]],"N":[[0,1,2],[-1,0,1],[-2,-1,0]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[0,-1,-2],[1,0,-1],[2,1,0]],\"N\":[[0,1,2],[-1,0,1],[-2,-1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0039","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]]、N=[[0, 1, 2, 3], [-1, 0, 1, 2], [-2, -1, 0, 1]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([3, 4], [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]], [[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1]], [[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1]])","explanation_zh":"有限結構核對=([3, 4], [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]], [[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1]], [[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162039,"parameters":{"M":[[1,0,-1,-2],[2,1,0,-1],[3,2,1,0]],"N":[[0,1,2,3],[-1,0,1,2],[-2,-1,0,1]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[1,0,-1,-2],[2,1,0,-1],[3,2,1,0]],\"N\":[[0,1,2,3],[-1,0,1,2],[-2,-1,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0040","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]]、N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3], [-2, -1, 0, 1, 2]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([3, 5], [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]])","explanation_zh":"有限結構核對=([3, 5], [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162040,"parameters":{"M":[[2,1,0,-1,-2],[3,2,1,0,-1],[4,3,2,1,0]],"N":[[0,1,2,3,4],[-1,0,1,2,3],[-2,-1,0,1,2]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[2,1,0,-1,-2],[3,2,1,0,-1],[4,3,2,1,0]],\"N\":[[0,1,2,3,4],[-1,0,1,2,3],[-2,-1,0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0041","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[3], [4], [5], [6]]、N=[[0], [-1], [-2], [-3]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([4, 1], [[3], [3], [3], [3]], [[3, 3, 3, 3]], [[3, 3, 3, 3]])","explanation_zh":"有限結構核對=([4, 1], [[3], [3], [3], [3]], [[3, 3, 3, 3]], [[3, 3, 3, 3]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162041,"parameters":{"M":[[3],[4],[5],[6]],"N":[[0],[-1],[-2],[-3]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[3],[4],[5],[6]],\"N\":[[0],[-1],[-2],[-3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0042","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[4, 3], [5, 4], [6, 5], [7, 6]]、N=[[0, 1], [-1, 0], [-2, -1], [-3, -2]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([4, 2], [[4, 4], [4, 4], [4, 4], [4, 4]], [[4, 4, 4, 4], [4, 4, 4, 4]], [[4, 4, 4, 4], [4, 4, 4, 4]])","explanation_zh":"有限結構核對=([4, 2], [[4, 4], [4, 4], [4, 4], [4, 4]], [[4, 4, 4, 4], [4, 4, 4, 4]], [[4, 4, 4, 4], [4, 4, 4, 4]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162042,"parameters":{"M":[[4,3],[5,4],[6,5],[7,6]],"N":[[0,1],[-1,0],[-2,-1],[-3,-2]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[4,3],[5,4],[6,5],[7,6]],\"N\":[[0,1],[-1,0],[-2,-1],[-3,-2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0043","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]]、N=[[0, 1, 2], [-1, 0, 1], [-2, -1, 0], [-3, -2, -1]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([4, 3], [[5, 5, 5], [5, 5, 5], [5, 5, 5], [5, 5, 5]], [[5, 5, 5, 5], [5, 5, 5, 5], [5, 5, 5, 5]], [[5, 5, 5, 5], [5, 5, 5, 5], [5, 5, 5, 5]])","explanation_zh":"有限結構核對=([4, 3], [[5, 5, 5], [5, 5, 5], [5, 5, 5], [5, 5, 5]], [[5, 5, 5, 5], [5, 5, 5, 5], [5, 5, 5, 5]], [[5, 5, 5, 5], [5, 5, 5, 5], [5, 5, 5, 5]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162043,"parameters":{"M":[[5,4,3],[6,5,4],[7,6,5],[8,7,6]],"N":[[0,1,2],[-1,0,1],[-2,-1,0],[-3,-2,-1]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[5,4,3],[6,5,4],[7,6,5],[8,7,6]],\"N\":[[0,1,2],[-1,0,1],[-2,-1,0],[-3,-2,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0044","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]]、N=[[0, 1, 2, 3], [-1, 0, 1, 2], [-2, -1, 0, 1], [-3, -2, -1, 0]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([4, 4], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]])","explanation_zh":"有限結構核對=([4, 4], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162044,"parameters":{"M":[[6,5,4,3],[7,6,5,4],[8,7,6,5],[9,8,7,6]],"N":[[0,1,2,3],[-1,0,1,2],[-2,-1,0,1],[-3,-2,-1,0]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[6,5,4,3],[7,6,5,4],[8,7,6,5],[9,8,7,6]],\"N\":[[0,1,2,3],[-1,0,1,2],[-2,-1,0,1],[-3,-2,-1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0045","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]]、N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3], [-2, -1, 0, 1, 2], [-3, -2, -1, 0, 1]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([4, 5], [[7, 7, 7, 7, 7], [7, 7, 7, 7, 7], [7, 7, 7, 7, 7], [7, 7, 7, 7, 7]], [[7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7]], [[7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7]])","explanation_zh":"有限結構核對=([4, 5], [[7, 7, 7, 7, 7], [7, 7, 7, 7, 7], [7, 7, 7, 7, 7], [7, 7, 7, 7, 7]], [[7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7]], [[7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162045,"parameters":{"M":[[7,6,5,4,3],[8,7,6,5,4],[9,8,7,6,5],[10,9,8,7,6]],"N":[[0,1,2,3,4],[-1,0,1,2,3],[-2,-1,0,1,2],[-3,-2,-1,0,1]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[7,6,5,4,3],[8,7,6,5,4],[9,8,7,6,5],[10,9,8,7,6]],\"N\":[[0,1,2,3,4],[-1,0,1,2,3],[-2,-1,0,1,2],[-3,-2,-1,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0046","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[8], [9], [10], [11], [12]]、N=[[0], [-1], [-2], [-3], [-4]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([5, 1], [[8], [8], [8], [8], [8]], [[8, 8, 8, 8, 8]], [[8, 8, 8, 8, 8]])","explanation_zh":"有限結構核對=([5, 1], [[8], [8], [8], [8], [8]], [[8, 8, 8, 8, 8]], [[8, 8, 8, 8, 8]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162046,"parameters":{"M":[[8],[9],[10],[11],[12]],"N":[[0],[-1],[-2],[-3],[-4]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[8],[9],[10],[11],[12]],\"N\":[[0],[-1],[-2],[-3],[-4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0047","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]]、N=[[0, 1], [-1, 0], [-2, -1], [-3, -2], [-4, -3]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([5, 2], [[9, 9], [9, 9], [9, 9], [9, 9], [9, 9]], [[9, 9, 9, 9, 9], [9, 9, 9, 9, 9]], [[9, 9, 9, 9, 9], [9, 9, 9, 9, 9]])","explanation_zh":"有限結構核對=([5, 2], [[9, 9], [9, 9], [9, 9], [9, 9], [9, 9]], [[9, 9, 9, 9, 9], [9, 9, 9, 9, 9]], [[9, 9, 9, 9, 9], [9, 9, 9, 9, 9]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162047,"parameters":{"M":[[9,8],[10,9],[11,10],[12,11],[13,12]],"N":[[0,1],[-1,0],[-2,-1],[-3,-2],[-4,-3]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[9,8],[10,9],[11,10],[12,11],[13,12]],\"N\":[[0,1],[-1,0],[-2,-1],[-3,-2],[-4,-3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0048","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]]、N=[[0, 1, 2], [-1, 0, 1], [-2, -1, 0], [-3, -2, -1], [-4, -3, -2]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([5, 3], [[10, 10, 10], [10, 10, 10], [10, 10, 10], [10, 10, 10], [10, 10, 10]], [[10, 10, 10, 10, 10], [10, 10, 10, 10, 10], [10, 10, 10, 10, 10]], [[10, 10, 10, 10, 10], [10, 10, 10, 10, 10], [10, 10, 10, 10, 10]])","explanation_zh":"有限結構核對=([5, 3], [[10, 10, 10], [10, 10, 10], [10, 10, 10], [10, 10, 10], [10, 10, 10]], [[10, 10, 10, 10, 10], [10, 10, 10, 10, 10], [10, 10, 10, 10, 10]], [[10, 10, 10, 10, 10], [10, 10, 10, 10, 10], [10, 10, 10, 10, 10]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162048,"parameters":{"M":[[10,9,8],[11,10,9],[12,11,10],[13,12,11],[14,13,12]],"N":[[0,1,2],[-1,0,1],[-2,-1,0],[-3,-2,-1],[-4,-3,-2]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[10,9,8],[11,10,9],[12,11,10],[13,12,11],[14,13,12]],\"N\":[[0,1,2],[-1,0,1],[-2,-1,0],[-3,-2,-1],[-4,-3,-2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0049","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]]、N=[[0, 1, 2, 3], [-1, 0, 1, 2], [-2, -1, 0, 1], [-3, -2, -1, 0], [-4, -3, -2, -1]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([5, 4], [[11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11]], [[11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11]], [[11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11]])","explanation_zh":"有限結構核對=([5, 4], [[11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11]], [[11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11]], [[11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162049,"parameters":{"M":[[11,10,9,8],[12,11,10,9],[13,12,11,10],[14,13,12,11],[15,14,13,12]],"N":[[0,1,2,3],[-1,0,1,2],[-2,-1,0,1],[-3,-2,-1,0],[-4,-3,-2,-1]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[11,10,9,8],[12,11,10,9],[13,12,11,10],[14,13,12,11],[15,14,13,12]],\"N\":[[0,1,2,3],[-1,0,1,2],[-2,-1,0,1],[-3,-2,-1,0],[-4,-3,-2,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0050","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"同尺寸和的轉置兩路","stem_zh":"外部索引從1開始；每格完整整數資料。同尺寸M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]]、N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3], [-2, -1, 0, 1, 2], [-3, -2, -1, 0, 1], [-4, -3, -2, -1, 0]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。","answer":{"kind":"expression","value":"([5, 5], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]])","explanation_zh":"有限結構核對=([5, 5], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同尺寸和的轉置兩路"]},"variant":{"family_id":"qf-density16-matrix-sum-transpose","seed":202610162050,"parameters":{"M":[[12,11,10,9,8],[13,12,11,10,9],[14,13,12,11,10],[15,14,13,12,11],[16,15,14,13,12]],"N":[[0,1,2,3,4],[-1,0,1,2,3],[-2,-1,0,1,2],[-3,-2,-1,0,1],[-4,-3,-2,-1,0]],"domain":"Z_rect_1based","operation":"matrix-sum-transpose","semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[12,11,10,9,8],[13,12,11,10,9],[14,13,12,11,10],[15,14,13,12,11],[16,15,14,13,12]],\"N\":[[0,1,2,3,4],[-1,0,1,2,3],[-2,-1,0,1,2],[-3,-2,-1,0,1],[-4,-3,-2,-1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0051","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[0]]，原序(i,j)候選=[[1, 1], [1, 1], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([1, 1], [是, 是, 否], [0, 0, 不合索引])","explanation_zh":"有限結構核對=([1, 1], [是, 是, 否], [0, 0, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162051,"parameters":{"M":[[0]],"queries":[[1,1],[1,1],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,0],[1,1,0],[0,1,\"untyped\"]],\"shape\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0052","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-11, -12]]，原序(i,j)候選=[[1, 1], [1, 2], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([1, 2], [是, 是, 否], [-11, -12, 不合索引])","explanation_zh":"有限結構核對=([1, 2], [是, 是, 否], [-11, -12, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162052,"parameters":{"M":[[-11,-12]],"queries":[[1,1],[1,2],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-11],[1,2,-12],[0,1,\"untyped\"]],\"shape\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0053","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-10, -11, -12]]，原序(i,j)候選=[[1, 1], [1, 3], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([1, 3], [是, 是, 否], [-10, -12, 不合索引])","explanation_zh":"有限結構核對=([1, 3], [是, 是, 否], [-10, -12, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162053,"parameters":{"M":[[-10,-11,-12]],"queries":[[1,1],[1,3],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-10],[1,3,-12],[0,1,\"untyped\"]],\"shape\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0054","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-9, -10, -11, -12]]，原序(i,j)候選=[[1, 1], [1, 4], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([1, 4], [是, 是, 否], [-9, -12, 不合索引])","explanation_zh":"有限結構核對=([1, 4], [是, 是, 否], [-9, -12, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162054,"parameters":{"M":[[-9,-10,-11,-12]],"queries":[[1,1],[1,4],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-9],[1,4,-12],[0,1,\"untyped\"]],\"shape\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0055","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-8, -9, -10, -11, -12]]，原序(i,j)候選=[[1, 1], [1, 5], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([1, 5], [是, 是, 否], [-8, -12, 不合索引])","explanation_zh":"有限結構核對=([1, 5], [是, 是, 否], [-8, -12, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162055,"parameters":{"M":[[-8,-9,-10,-11,-12]],"queries":[[1,1],[1,5],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-8],[1,5,-12],[0,1,\"untyped\"]],\"shape\":[1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0056","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-7], [-6]]，原序(i,j)候選=[[1, 1], [2, 1], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([2, 1], [是, 是, 否], [-7, -6, 不合索引])","explanation_zh":"有限結構核對=([2, 1], [是, 是, 否], [-7, -6, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162056,"parameters":{"M":[[-7],[-6]],"queries":[[1,1],[2,1],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-7],[2,1,-6],[0,1,\"untyped\"]],\"shape\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0057","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-6, -7], [-5, -6]]，原序(i,j)候選=[[1, 1], [2, 2], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([2, 2], [是, 是, 否], [-6, -6, 不合索引])","explanation_zh":"有限結構核對=([2, 2], [是, 是, 否], [-6, -6, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162057,"parameters":{"M":[[-6,-7],[-5,-6]],"queries":[[1,1],[2,2],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-6],[2,2,-6],[0,1,\"untyped\"]],\"shape\":[2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0058","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-5, -6, -7], [-4, -5, -6]]，原序(i,j)候選=[[1, 1], [2, 3], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([2, 3], [是, 是, 否], [-5, -6, 不合索引])","explanation_zh":"有限結構核對=([2, 3], [是, 是, 否], [-5, -6, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162058,"parameters":{"M":[[-5,-6,-7],[-4,-5,-6]],"queries":[[1,1],[2,3],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-5],[2,3,-6],[0,1,\"untyped\"]],\"shape\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0059","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-4, -5, -6, -7], [-3, -4, -5, -6]]，原序(i,j)候選=[[1, 1], [2, 4], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([2, 4], [是, 是, 否], [-4, -6, 不合索引])","explanation_zh":"有限結構核對=([2, 4], [是, 是, 否], [-4, -6, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162059,"parameters":{"M":[[-4,-5,-6,-7],[-3,-4,-5,-6]],"queries":[[1,1],[2,4],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-4],[2,4,-6],[0,1,\"untyped\"]],\"shape\":[2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0060","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]]，原序(i,j)候選=[[1, 1], [2, 5], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([2, 5], [是, 是, 否], [-3, -6, 不合索引])","explanation_zh":"有限結構核對=([2, 5], [是, 是, 否], [-3, -6, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162060,"parameters":{"M":[[-3,-4,-5,-6,-7],[-2,-3,-4,-5,-6]],"queries":[[1,1],[2,5],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-3],[2,5,-6],[0,1,\"untyped\"]],\"shape\":[2,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0061","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-2], [-1], [0]]，原序(i,j)候選=[[1, 1], [3, 1], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([3, 1], [是, 是, 否], [-2, 0, 不合索引])","explanation_zh":"有限結構核對=([3, 1], [是, 是, 否], [-2, 0, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162061,"parameters":{"M":[[-2],[-1],[0]],"queries":[[1,1],[3,1],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-2],[3,1,0],[0,1,\"untyped\"]],\"shape\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0062","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-1, -2], [0, -1], [1, 0]]，原序(i,j)候選=[[1, 1], [3, 2], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([3, 2], [是, 是, 否], [-1, 0, 不合索引])","explanation_zh":"有限結構核對=([3, 2], [是, 是, 否], [-1, 0, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162062,"parameters":{"M":[[-1,-2],[0,-1],[1,0]],"queries":[[1,1],[3,2],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-1],[3,2,0],[0,1,\"untyped\"]],\"shape\":[3,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0063","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[0, -1, -2], [1, 0, -1], [2, 1, 0]]，原序(i,j)候選=[[1, 1], [3, 3], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([3, 3], [是, 是, 否], [0, 0, 不合索引])","explanation_zh":"有限結構核對=([3, 3], [是, 是, 否], [0, 0, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162063,"parameters":{"M":[[0,-1,-2],[1,0,-1],[2,1,0]],"queries":[[1,1],[3,3],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,0],[3,3,0],[0,1,\"untyped\"]],\"shape\":[3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0064","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]]，原序(i,j)候選=[[1, 1], [3, 4], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([3, 4], [是, 是, 否], [1, 0, 不合索引])","explanation_zh":"有限結構核對=([3, 4], [是, 是, 否], [1, 0, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162064,"parameters":{"M":[[1,0,-1,-2],[2,1,0,-1],[3,2,1,0]],"queries":[[1,1],[3,4],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,1],[3,4,0],[0,1,\"untyped\"]],\"shape\":[3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0065","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]]，原序(i,j)候選=[[1, 1], [3, 5], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([3, 5], [是, 是, 否], [2, 0, 不合索引])","explanation_zh":"有限結構核對=([3, 5], [是, 是, 否], [2, 0, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162065,"parameters":{"M":[[2,1,0,-1,-2],[3,2,1,0,-1],[4,3,2,1,0]],"queries":[[1,1],[3,5],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,2],[3,5,0],[0,1,\"untyped\"]],\"shape\":[3,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0066","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[3], [4], [5], [6]]，原序(i,j)候選=[[1, 1], [4, 1], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([4, 1], [是, 是, 否], [3, 6, 不合索引])","explanation_zh":"有限結構核對=([4, 1], [是, 是, 否], [3, 6, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162066,"parameters":{"M":[[3],[4],[5],[6]],"queries":[[1,1],[4,1],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,3],[4,1,6],[0,1,\"untyped\"]],\"shape\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0067","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[4, 3], [5, 4], [6, 5], [7, 6]]，原序(i,j)候選=[[1, 1], [4, 2], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([4, 2], [是, 是, 否], [4, 6, 不合索引])","explanation_zh":"有限結構核對=([4, 2], [是, 是, 否], [4, 6, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162067,"parameters":{"M":[[4,3],[5,4],[6,5],[7,6]],"queries":[[1,1],[4,2],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,4],[4,2,6],[0,1,\"untyped\"]],\"shape\":[4,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0068","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]]，原序(i,j)候選=[[1, 1], [4, 3], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([4, 3], [是, 是, 否], [5, 6, 不合索引])","explanation_zh":"有限結構核對=([4, 3], [是, 是, 否], [5, 6, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162068,"parameters":{"M":[[5,4,3],[6,5,4],[7,6,5],[8,7,6]],"queries":[[1,1],[4,3],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,5],[4,3,6],[0,1,\"untyped\"]],\"shape\":[4,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0069","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]]，原序(i,j)候選=[[1, 1], [4, 4], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([4, 4], [是, 是, 否], [6, 6, 不合索引])","explanation_zh":"有限結構核對=([4, 4], [是, 是, 否], [6, 6, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162069,"parameters":{"M":[[6,5,4,3],[7,6,5,4],[8,7,6,5],[9,8,7,6]],"queries":[[1,1],[4,4],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,6],[4,4,6],[0,1,\"untyped\"]],\"shape\":[4,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0070","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]]，原序(i,j)候選=[[1, 1], [4, 5], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([4, 5], [是, 是, 否], [7, 6, 不合索引])","explanation_zh":"有限結構核對=([4, 5], [是, 是, 否], [7, 6, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162070,"parameters":{"M":[[7,6,5,4,3],[8,7,6,5,4],[9,8,7,6,5],[10,9,8,7,6]],"queries":[[1,1],[4,5],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,7],[4,5,6],[0,1,\"untyped\"]],\"shape\":[4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0071","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[8], [9], [10], [11], [12]]，原序(i,j)候選=[[1, 1], [5, 1], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([5, 1], [是, 是, 否], [8, 12, 不合索引])","explanation_zh":"有限結構核對=([5, 1], [是, 是, 否], [8, 12, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162071,"parameters":{"M":[[8],[9],[10],[11],[12]],"queries":[[1,1],[5,1],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,8],[5,1,12],[0,1,\"untyped\"]],\"shape\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0072","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]]，原序(i,j)候選=[[1, 1], [5, 2], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([5, 2], [是, 是, 否], [9, 12, 不合索引])","explanation_zh":"有限結構核對=([5, 2], [是, 是, 否], [9, 12, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162072,"parameters":{"M":[[9,8],[10,9],[11,10],[12,11],[13,12]],"queries":[[1,1],[5,2],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,9],[5,2,12],[0,1,\"untyped\"]],\"shape\":[5,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0073","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]]，原序(i,j)候選=[[1, 1], [5, 3], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([5, 3], [是, 是, 否], [10, 12, 不合索引])","explanation_zh":"有限結構核對=([5, 3], [是, 是, 否], [10, 12, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162073,"parameters":{"M":[[10,9,8],[11,10,9],[12,11,10],[13,12,11],[14,13,12]],"queries":[[1,1],[5,3],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,10],[5,3,12],[0,1,\"untyped\"]],\"shape\":[5,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0074","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]]，原序(i,j)候選=[[1, 1], [5, 4], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([5, 4], [是, 是, 否], [11, 12, 不合索引])","explanation_zh":"有限結構核對=([5, 4], [是, 是, 否], [11, 12, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162074,"parameters":{"M":[[11,10,9,8],[12,11,10,9],[13,12,11,10],[14,13,12,11],[15,14,13,12]],"queries":[[1,1],[5,4],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,11],[5,4,12],[0,1,\"untyped\"]],\"shape\":[5,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0075","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"三個外部索引候選","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]]，原序(i,j)候選=[[1, 1], [5, 5], [0, 1]]。求 (M尺寸, 三個索引資格, 三個值或不合索引)。不合索引時不讀該格。","answer":{"kind":"expression","value":"([5, 5], [是, 是, 否], [12, 12, 不合索引])","explanation_zh":"有限結構核對=([5, 5], [是, 是, 否], [12, 12, 不合索引])。兩個索引都在1到尺寸內才讀元素，其他格不充作本題身份。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個外部索引候選"]},"variant":{"family_id":"qf-density16-matrix-three-index-qualification","seed":202610162075,"parameters":{"M":[[12,11,10,9,8],[13,12,11,10,9],[14,13,12,11,10],[15,14,13,12,11],[16,15,14,13,12]],"queries":[[1,1],[5,5],[0,1]],"domain":"Z_rect_1based","operation":"matrix-three-index-qualification","semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,12],[5,5,12],[0,1,\"untyped\"]],\"shape\":[5,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0076","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[0]]，候選矩陣=[[0]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[0]], [1, 1], 是, [[0]], 是)","explanation_zh":"有限結構核對=([[0]], [1, 1], 是, [[0]], 是)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162076,"parameters":{"M":[[0]],"candidate":[[0]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[0]],\"candidate\":[[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0077","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[1, 2]]，候選矩陣=[[1, 2]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[1], [2]], [1, 2], 否, 尺寸不合, 否)","explanation_zh":"有限結構核對=([[1], [2]], [1, 2], 否, 尺寸不合, 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162077,"parameters":{"M":[[1,2]],"candidate":[[1,2]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[1,2]],\"candidate\":[[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0078","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[1, 2], [3, 4]]，候選矩陣=[[1, 3], [2, 4]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[1, 3], [2, 4]], [2, 2], 是, [[0, 0], [0, 0]], 是)","explanation_zh":"有限結構核對=([[1, 3], [2, 4]], [2, 2], 是, [[0, 0], [0, 0]], 是)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162078,"parameters":{"M":[[1,2],[3,4]],"candidate":[[1,3],[2,4]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[1,2],[3,4]],\"candidate\":[[1,3],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0079","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[1, 2], [3, 4]]，候選矩陣=[[1, 0], [2, 4]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[1, 3], [2, 4]], [2, 2], 是, [[0, -3], [0, 0]], 否)","explanation_zh":"有限結構核對=([[1, 3], [2, 4]], [2, 2], 是, [[0, -3], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162079,"parameters":{"M":[[1,2],[3,4]],"candidate":[[1,0],[2,4]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[1,2],[3,4]],\"candidate\":[[1,0],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0080","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-8, 1], [2, -1]]，候選矩陣=[[-7, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[-8, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[-8, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162080,"parameters":{"M":[[-8,1],[2,-1]],"candidate":[[-7,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-8,1],[2,-1]],\"candidate\":[[-7,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0081","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-7, 1], [2, -1]]，候選矩陣=[[-6, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[-7, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[-7, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162081,"parameters":{"M":[[-7,1],[2,-1]],"candidate":[[-6,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-7,1],[2,-1]],\"candidate\":[[-6,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0082","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-6, 1], [2, -1]]，候選矩陣=[[-5, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[-6, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[-6, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162082,"parameters":{"M":[[-6,1],[2,-1]],"candidate":[[-5,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-6,1],[2,-1]],\"candidate\":[[-5,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0083","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-5, 1], [2, -1]]，候選矩陣=[[-4, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[-5, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[-5, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162083,"parameters":{"M":[[-5,1],[2,-1]],"candidate":[[-4,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-5,1],[2,-1]],\"candidate\":[[-4,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0084","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-4, 1], [2, -1]]，候選矩陣=[[-3, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[-4, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[-4, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162084,"parameters":{"M":[[-4,1],[2,-1]],"candidate":[[-3,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-4,1],[2,-1]],\"candidate\":[[-3,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0085","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-3, 1], [2, -1]]，候選矩陣=[[-2, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[-3, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[-3, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162085,"parameters":{"M":[[-3,1],[2,-1]],"candidate":[[-2,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-3,1],[2,-1]],\"candidate\":[[-2,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0086","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-2, 1], [2, -1]]，候選矩陣=[[-1, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[-2, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[-2, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162086,"parameters":{"M":[[-2,1],[2,-1]],"candidate":[[-1,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-2,1],[2,-1]],\"candidate\":[[-1,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0087","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[-1, 1], [2, -1]]，候選矩陣=[[0, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[-1, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[-1, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162087,"parameters":{"M":[[-1,1],[2,-1]],"candidate":[[0,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-1,1],[2,-1]],\"candidate\":[[0,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0088","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[0, 1], [2, -1]]，候選矩陣=[[1, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[0, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[0, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162088,"parameters":{"M":[[0,1],[2,-1]],"candidate":[[1,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[0,1],[2,-1]],\"candidate\":[[1,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0089","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[1, 1], [2, -1]]，候選矩陣=[[2, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[1, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[1, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162089,"parameters":{"M":[[1,1],[2,-1]],"candidate":[[2,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[1,1],[2,-1]],\"candidate\":[[2,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0090","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[2, 1], [2, -1]]，候選矩陣=[[3, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[2, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[2, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162090,"parameters":{"M":[[2,1],[2,-1]],"candidate":[[3,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[2,1],[2,-1]],\"candidate\":[[3,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0091","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[3, 1], [2, -1]]，候選矩陣=[[4, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[3, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[3, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162091,"parameters":{"M":[[3,1],[2,-1]],"candidate":[[4,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[3,1],[2,-1]],\"candidate\":[[4,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0092","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[4, 1], [2, -1]]，候選矩陣=[[5, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[4, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[4, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162092,"parameters":{"M":[[4,1],[2,-1]],"candidate":[[5,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[4,1],[2,-1]],\"candidate\":[[5,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0093","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[5, 1], [2, -1]]，候選矩陣=[[6, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[5, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[5, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162093,"parameters":{"M":[[5,1],[2,-1]],"candidate":[[6,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[5,1],[2,-1]],\"candidate\":[[6,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0094","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[6, 1], [2, -1]]，候選矩陣=[[7, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[6, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[6, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162094,"parameters":{"M":[[6,1],[2,-1]],"candidate":[[7,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[6,1],[2,-1]],\"candidate\":[[7,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0095","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[7, 1], [2, -1]]，候選矩陣=[[8, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[7, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[7, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162095,"parameters":{"M":[[7,1],[2,-1]],"candidate":[[8,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[7,1],[2,-1]],\"candidate\":[[8,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0096","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[8, 1], [2, -1]]，候選矩陣=[[9, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[8, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[8, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162096,"parameters":{"M":[[8,1],[2,-1]],"candidate":[[9,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[8,1],[2,-1]],\"candidate\":[[9,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0097","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[9, 1], [2, -1]]，候選矩陣=[[10, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[9, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[9, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162097,"parameters":{"M":[[9,1],[2,-1]],"candidate":[[10,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[9,1],[2,-1]],\"candidate\":[[10,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0098","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[10, 1], [2, -1]]，候選矩陣=[[11, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[10, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[10, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162098,"parameters":{"M":[[10,1],[2,-1]],"candidate":[[11,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[10,1],[2,-1]],\"candidate\":[[11,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0099","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[11, 1], [2, -1]]，候選矩陣=[[12, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[11, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[11, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162099,"parameters":{"M":[[11,1],[2,-1]],"candidate":[[12,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[11,1],[2,-1]],\"candidate\":[[12,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r02-matrix-0100","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","language":"zh-Hant","source_mko_ids":["mko-matrix"],"title_zh":"轉置候選的尺寸與殘差","stem_zh":"外部索引從1開始；每格完整整數資料。M=[[12, 1], [2, -1]]，候選矩陣=[[13, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。","answer":{"kind":"expression","value":"([[12, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)","explanation_zh":"有限結構核對=([[12, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["轉置候選的尺寸與殘差"]},"variant":{"family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162100,"parameters":{"M":[[12,1],[2,-1]],"candidate":[[13,2],[1,-1]],"domain":"Z_rect_1based","operation":"matrix-candidate-transpose-audit","semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[12,1],[2,-1]],\"candidate\":[[13,2],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}