{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r01-bijective-0001","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(0)=∅→Fin(0)=∅；總函數f依輸入索引列images=[]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([], [], [], [], 是, 是, 是)","explanation_zh":"離散結構核對=([], [], [], [], 是, 是, 是)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171001,"parameters":{"n":0,"m":0,"images":[],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[],\"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-20261017-r01-bijective-0002","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(0)=∅→Fin(2)=[0, 1]；總函數f依輸入索引列images=[]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([], [[], []], [], [0, 1], 是, 否, 否)","explanation_zh":"離散結構核對=([], [[], []], [], [0, 1], 是, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171002,"parameters":{"n":0,"m":2,"images":[],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[],\"m\":2,\"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-20261017-r01-bijective-0003","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(1)=[0]→Fin(1)=[0]；總函數f依輸入索引列images=[0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([0], [[0]], [], [], 是, 是, 是)","explanation_zh":"離散結構核對=([0], [[0]], [], [], 是, 是, 是)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171003,"parameters":{"n":1,"m":1,"images":[0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[0],\"m\":1,\"n\":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-20261017-r01-bijective-0004","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(3)=[0, 1, 2]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1, 2]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([0, 1, 2], [[0], [1], [2]], [], [], 是, 是, 是)","explanation_zh":"離散結構核對=([0, 1, 2], [[0], [1], [2]], [], [], 是, 是, 是)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171004,"parameters":{"n":3,"m":3,"images":[0,1,2],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[0,1,2],\"m\":3,\"n\":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-20261017-r01-bijective-0005","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(3)=[0, 1, 2]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 0, 1]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([0, 0, 1], [[0, 1], [2], []], [[0, 1]], [2], 否, 否, 否)","explanation_zh":"離散結構核對=([0, 0, 1], [[0, 1], [2], []], [[0, 1]], [2], 否, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171005,"parameters":{"n":3,"m":3,"images":[0,0,1],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[0,0,1],\"m\":3,\"n\":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-20261017-r01-bijective-0006","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(3)=[0, 1, 2]→Fin(2)=[0, 1]；總函數f依輸入索引列images=[0, 1, 1]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([0, 1, 1], [[0], [1, 2]], [[1, 2]], [], 否, 是, 否)","explanation_zh":"離散結構核對=([0, 1, 1], [[0], [1, 2]], [[1, 2]], [], 否, 是, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171006,"parameters":{"n":3,"m":2,"images":[0,1,1],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[0,1,1],\"m\":2,\"n\":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-20261017-r01-bijective-0007","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(2)=[0, 1]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([0, 1], [[0], [1], []], [], [2], 是, 否, 否)","explanation_zh":"離散結構核對=([0, 1], [[0], [1], []], [], [2], 是, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171007,"parameters":{"n":2,"m":3,"images":[0,1],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[0,1],\"m\":3,\"n\":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-20261017-r01-bijective-0008","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 0, 0, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([0, 0, 0, 0, 0], [[0, 1, 2, 3, 4], [], []], [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], [1, 2], 否, 否, 否)","explanation_zh":"離散結構核對=([0, 0, 0, 0, 0], [[0, 1, 2, 3, 4], [], []], [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], [1, 2], 否, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171008,"parameters":{"n":5,"m":3,"images":[0,0,0,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[0,0,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0009","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 0, 0, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([1, 0, 0, 0, 0], [[1, 2, 3, 4], [0], []], [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], [2], 否, 否, 否)","explanation_zh":"離散結構核對=([1, 0, 0, 0, 0], [[1, 2, 3, 4], [0], []], [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], [2], 否, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171009,"parameters":{"n":5,"m":3,"images":[1,0,0,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[1,0,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0010","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 0, 0, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([2, 0, 0, 0, 0], [[1, 2, 3, 4], [], [0]], [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], [1], 否, 否, 否)","explanation_zh":"離散結構核對=([2, 0, 0, 0, 0], [[1, 2, 3, 4], [], [0]], [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], [1], 否, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171010,"parameters":{"n":5,"m":3,"images":[2,0,0,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[2,0,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0011","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1, 0, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([0, 1, 0, 0, 0], [[0, 2, 3, 4], [1], []], [[0, 2], [0, 3], [0, 4], [2, 3], [2, 4], [3, 4]], [2], 否, 否, 否)","explanation_zh":"離散結構核對=([0, 1, 0, 0, 0], [[0, 2, 3, 4], [1], []], [[0, 2], [0, 3], [0, 4], [2, 3], [2, 4], [3, 4]], [2], 否, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171011,"parameters":{"n":5,"m":3,"images":[0,1,0,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[0,1,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0012","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 1, 0, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([1, 1, 0, 0, 0], [[2, 3, 4], [0, 1], []], [[0, 1], [2, 3], [2, 4], [3, 4]], [2], 否, 否, 否)","explanation_zh":"離散結構核對=([1, 1, 0, 0, 0], [[2, 3, 4], [0, 1], []], [[0, 1], [2, 3], [2, 4], [3, 4]], [2], 否, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171012,"parameters":{"n":5,"m":3,"images":[1,1,0,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[1,1,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0013","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 1, 0, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], [[2, 3, 4], [1], [0]], [[2, 3], [2, 4], [3, 4]], [], 否, 是, 否)","explanation_zh":"離散結構核對=([2, 1, 0, 0, 0], [[2, 3, 4], [1], [0]], [[2, 3], [2, 4], [3, 4]], [], 否, 是, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171013,"parameters":{"n":5,"m":3,"images":[2,1,0,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[2,1,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0014","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 2, 0, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([0, 2, 0, 0, 0], [[0, 2, 3, 4], [], [1]], [[0, 2], [0, 3], [0, 4], [2, 3], [2, 4], [3, 4]], [1], 否, 否, 否)","explanation_zh":"離散結構核對=([0, 2, 0, 0, 0], [[0, 2, 3, 4], [], [1]], [[0, 2], [0, 3], [0, 4], [2, 3], [2, 4], [3, 4]], [1], 否, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171014,"parameters":{"n":5,"m":3,"images":[0,2,0,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[0,2,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0015","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 2, 0, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([1, 2, 0, 0, 0], [[2, 3, 4], [0], [1]], [[2, 3], [2, 4], [3, 4]], [], 否, 是, 否)","explanation_zh":"離散結構核對=([1, 2, 0, 0, 0], [[2, 3, 4], [0], [1]], [[2, 3], [2, 4], [3, 4]], [], 否, 是, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171015,"parameters":{"n":5,"m":3,"images":[1,2,0,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[1,2,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0016","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 2, 0, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([2, 2, 0, 0, 0], [[2, 3, 4], [], [0, 1]], [[0, 1], [2, 3], [2, 4], [3, 4]], [1], 否, 否, 否)","explanation_zh":"離散結構核對=([2, 2, 0, 0, 0], [[2, 3, 4], [], [0, 1]], [[0, 1], [2, 3], [2, 4], [3, 4]], [1], 否, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171016,"parameters":{"n":5,"m":3,"images":[2,2,0,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[2,2,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0017","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 0, 1, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([0, 0, 1, 0, 0], [[0, 1, 3, 4], [2], []], [[0, 1], [0, 3], [0, 4], [1, 3], [1, 4], [3, 4]], [2], 否, 否, 否)","explanation_zh":"離散結構核對=([0, 0, 1, 0, 0], [[0, 1, 3, 4], [2], []], [[0, 1], [0, 3], [0, 4], [1, 3], [1, 4], [3, 4]], [2], 否, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171017,"parameters":{"n":5,"m":3,"images":[0,0,1,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[0,0,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0018","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 0, 1, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([1, 0, 1, 0, 0], [[1, 3, 4], [0, 2], []], [[0, 2], [1, 3], [1, 4], [3, 4]], [2], 否, 否, 否)","explanation_zh":"離散結構核對=([1, 0, 1, 0, 0], [[1, 3, 4], [0, 2], []], [[0, 2], [1, 3], [1, 4], [3, 4]], [2], 否, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171018,"parameters":{"n":5,"m":3,"images":[1,0,1,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[1,0,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0019","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 0, 1, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([2, 0, 1, 0, 0], [[1, 3, 4], [2], [0]], [[1, 3], [1, 4], [3, 4]], [], 否, 是, 否)","explanation_zh":"離散結構核對=([2, 0, 1, 0, 0], [[1, 3, 4], [2], [0]], [[1, 3], [1, 4], [3, 4]], [], 否, 是, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171019,"parameters":{"n":5,"m":3,"images":[2,0,1,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[2,0,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0020","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1, 1, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([0, 1, 1, 0, 0], [[0, 3, 4], [1, 2], []], [[0, 3], [0, 4], [1, 2], [3, 4]], [2], 否, 否, 否)","explanation_zh":"離散結構核對=([0, 1, 1, 0, 0], [[0, 3, 4], [1, 2], []], [[0, 3], [0, 4], [1, 2], [3, 4]], [2], 否, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171020,"parameters":{"n":5,"m":3,"images":[0,1,1,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[0,1,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0021","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 1, 1, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([1, 1, 1, 0, 0], [[3, 4], [0, 1, 2], []], [[0, 1], [0, 2], [1, 2], [3, 4]], [2], 否, 否, 否)","explanation_zh":"離散結構核對=([1, 1, 1, 0, 0], [[3, 4], [0, 1, 2], []], [[0, 1], [0, 2], [1, 2], [3, 4]], [2], 否, 否, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171021,"parameters":{"n":5,"m":3,"images":[1,1,1,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[1,1,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0022","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 1, 1, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([2, 1, 1, 0, 0], [[3, 4], [1, 2], [0]], [[1, 2], [3, 4]], [], 否, 是, 否)","explanation_zh":"離散結構核對=([2, 1, 1, 0, 0], [[3, 4], [1, 2], [0]], [[1, 2], [3, 4]], [], 否, 是, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171022,"parameters":{"n":5,"m":3,"images":[2,1,1,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[2,1,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0023","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 2, 1, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([0, 2, 1, 0, 0], [[0, 3, 4], [2], [1]], [[0, 3], [0, 4], [3, 4]], [], 否, 是, 否)","explanation_zh":"離散結構核對=([0, 2, 1, 0, 0], [[0, 3, 4], [2], [1]], [[0, 3], [0, 4], [3, 4]], [], 否, 是, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171023,"parameters":{"n":5,"m":3,"images":[0,2,1,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[0,2,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0024","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 2, 1, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([1, 2, 1, 0, 0], [[3, 4], [0, 2], [1]], [[0, 2], [3, 4]], [], 否, 是, 否)","explanation_zh":"離散結構核對=([1, 2, 1, 0, 0], [[3, 4], [0, 2], [1]], [[0, 2], [3, 4]], [], 否, 是, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171024,"parameters":{"n":5,"m":3,"images":[1,2,1,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[1,2,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0025","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"完整纖維與三種映射資格","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 2, 1, 0, 0]。依序答 (images, 全陪域纖維, 所有相撞輸入對, 缺失目標, 單射, 滿射, 雙射)。纖維按目標索引列齊，空纖維也保留；相撞對按字典序。","answer":{"kind":"expression","value":"([2, 2, 1, 0, 0], [[3, 4], [2], [0, 1]], [[0, 1], [3, 4]], [], 否, 是, 否)","explanation_zh":"離散結構核對=([2, 2, 1, 0, 0], [[3, 4], [2], [0, 1]], [[0, 1], [3, 4]], [], 否, 是, 否)；逐個輸入放進其目標纖維，再列所有相同輸出的相異輸入對。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整纖維與三種映射資格"]},"variant":{"family_id":"qf-density17-bij-fibers-and-class","seed":202610171025,"parameters":{"n":5,"m":3,"images":[2,2,1,0,0],"domain":"Fin_total_table","operation":"bij-fibers-and-class","semantic_template":"qf-density17-bij-fibers-and-class/{\"images\":[2,2,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0026","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(0)=∅→Fin(0)=∅；總函數f依輸入索引列images=[]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([], [], 是, [])","explanation_zh":"離散結構核對=([], [], 是, [])；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171026,"parameters":{"n":0,"m":0,"images":[],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[],\"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-20261017-r01-bijective-0027","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(0)=∅→Fin(2)=[0, 1]；總函數f依輸入索引列images=[]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([], [0, 0], 否, 不成反函數)","explanation_zh":"離散結構核對=([], [0, 0], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171027,"parameters":{"n":0,"m":2,"images":[],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[],\"m\":2,\"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-20261017-r01-bijective-0028","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(1)=[0]→Fin(1)=[0]；總函數f依輸入索引列images=[0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 0]], [1], 是, [0])","explanation_zh":"離散結構核對=([[0, 0]], [1], 是, [0])；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171028,"parameters":{"n":1,"m":1,"images":[0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[0],\"m\":1,\"n\":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-20261017-r01-bijective-0029","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(3)=[0, 1, 2]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1, 2]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 0], [1, 1], [2, 2]], [1, 1, 1], 是, [0, 1, 2])","explanation_zh":"離散結構核對=([[0, 0], [1, 1], [2, 2]], [1, 1, 1], 是, [0, 1, 2])；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171029,"parameters":{"n":3,"m":3,"images":[0,1,2],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[0,1,2],\"m\":3,\"n\":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-20261017-r01-bijective-0030","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(3)=[0, 1, 2]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 0, 1]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 0], [0, 1], [1, 2]], [2, 1, 0], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 0], [0, 1], [1, 2]], [2, 1, 0], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171030,"parameters":{"n":3,"m":3,"images":[0,0,1],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[0,0,1],\"m\":3,\"n\":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-20261017-r01-bijective-0031","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(3)=[0, 1, 2]→Fin(2)=[0, 1]；總函數f依輸入索引列images=[0, 1, 1]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 0], [1, 1], [1, 2]], [1, 2], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 0], [1, 1], [1, 2]], [1, 2], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171031,"parameters":{"n":3,"m":2,"images":[0,1,1],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[0,1,1],\"m\":2,\"n\":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-20261017-r01-bijective-0032","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(2)=[0, 1]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 0], [1, 1]], [1, 1, 0], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 0], [1, 1]], [1, 1, 0], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171032,"parameters":{"n":2,"m":3,"images":[0,1],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[0,1],\"m\":3,\"n\":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-20261017-r01-bijective-0033","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 0, 0, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 0], [0, 1], [0, 2], [0, 3], [0, 4]], [5, 0, 0], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 0], [0, 1], [0, 2], [0, 3], [0, 4]], [5, 0, 0], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171033,"parameters":{"n":5,"m":3,"images":[0,0,0,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[0,0,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0034","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 0, 0, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 3], [0, 4], [1, 0]], [4, 1, 0], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 1], [0, 2], [0, 3], [0, 4], [1, 0]], [4, 1, 0], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171034,"parameters":{"n":5,"m":3,"images":[1,0,0,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[1,0,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0035","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 0, 0, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 3], [0, 4], [2, 0]], [4, 0, 1], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 1], [0, 2], [0, 3], [0, 4], [2, 0]], [4, 0, 1], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171035,"parameters":{"n":5,"m":3,"images":[2,0,0,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[2,0,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0036","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1, 0, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 0], [0, 2], [0, 3], [0, 4], [1, 1]], [4, 1, 0], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 0], [0, 2], [0, 3], [0, 4], [1, 1]], [4, 1, 0], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171036,"parameters":{"n":5,"m":3,"images":[0,1,0,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[0,1,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0037","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 1, 0, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 2], [0, 3], [0, 4], [1, 0], [1, 1]], [3, 2, 0], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 2], [0, 3], [0, 4], [1, 0], [1, 1]], [3, 2, 0], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171037,"parameters":{"n":5,"m":3,"images":[1,1,0,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[1,1,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0038","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 1, 0, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 2], [0, 3], [0, 4], [1, 1], [2, 0]], [3, 1, 1], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 2], [0, 3], [0, 4], [1, 1], [2, 0]], [3, 1, 1], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171038,"parameters":{"n":5,"m":3,"images":[2,1,0,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[2,1,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0039","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 2, 0, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 0], [0, 2], [0, 3], [0, 4], [2, 1]], [4, 0, 1], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 0], [0, 2], [0, 3], [0, 4], [2, 1]], [4, 0, 1], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171039,"parameters":{"n":5,"m":3,"images":[0,2,0,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[0,2,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0040","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 2, 0, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 2], [0, 3], [0, 4], [1, 0], [2, 1]], [3, 1, 1], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 2], [0, 3], [0, 4], [1, 0], [2, 1]], [3, 1, 1], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171040,"parameters":{"n":5,"m":3,"images":[1,2,0,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[1,2,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0041","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 2, 0, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 2], [0, 3], [0, 4], [2, 0], [2, 1]], [3, 0, 2], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 2], [0, 3], [0, 4], [2, 0], [2, 1]], [3, 0, 2], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171041,"parameters":{"n":5,"m":3,"images":[2,2,0,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[2,2,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0042","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 0, 1, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 0], [0, 1], [0, 3], [0, 4], [1, 2]], [4, 1, 0], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 0], [0, 1], [0, 3], [0, 4], [1, 2]], [4, 1, 0], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171042,"parameters":{"n":5,"m":3,"images":[0,0,1,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[0,0,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0043","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 0, 1, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 1], [0, 3], [0, 4], [1, 0], [1, 2]], [3, 2, 0], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 1], [0, 3], [0, 4], [1, 0], [1, 2]], [3, 2, 0], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171043,"parameters":{"n":5,"m":3,"images":[1,0,1,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[1,0,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0044","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 0, 1, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 1], [0, 3], [0, 4], [1, 2], [2, 0]], [3, 1, 1], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 1], [0, 3], [0, 4], [1, 2], [2, 0]], [3, 1, 1], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171044,"parameters":{"n":5,"m":3,"images":[2,0,1,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[2,0,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0045","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1, 1, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 0], [0, 3], [0, 4], [1, 1], [1, 2]], [3, 2, 0], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 0], [0, 3], [0, 4], [1, 1], [1, 2]], [3, 2, 0], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171045,"parameters":{"n":5,"m":3,"images":[0,1,1,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[0,1,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0046","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 1, 1, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 3], [0, 4], [1, 0], [1, 1], [1, 2]], [2, 3, 0], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 3], [0, 4], [1, 0], [1, 1], [1, 2]], [2, 3, 0], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171046,"parameters":{"n":5,"m":3,"images":[1,1,1,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[1,1,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0047","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 1, 1, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 3], [0, 4], [1, 1], [1, 2], [2, 0]], [2, 2, 1], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 3], [0, 4], [1, 1], [1, 2], [2, 0]], [2, 2, 1], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171047,"parameters":{"n":5,"m":3,"images":[2,1,1,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[2,1,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0048","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 2, 1, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 0], [0, 3], [0, 4], [1, 2], [2, 1]], [3, 1, 1], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 0], [0, 3], [0, 4], [1, 2], [2, 1]], [3, 1, 1], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171048,"parameters":{"n":5,"m":3,"images":[0,2,1,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[0,2,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0049","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 2, 1, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 3], [0, 4], [1, 0], [1, 2], [2, 1]], [2, 2, 1], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 3], [0, 4], [1, 0], [1, 2], [2, 1]], [2, 2, 1], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171049,"parameters":{"n":5,"m":3,"images":[1,2,1,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[1,2,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0050","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換關係與反函數門檻","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 2, 1, 0, 0]。求 (交換後有序對字典序列, 全目標纖維大小, 雙射資格, 反函數總表或不成反函數)。","answer":{"kind":"expression","value":"([[0, 3], [0, 4], [1, 2], [2, 0], [2, 1]], [2, 1, 2], 否, 不成反函數)","explanation_zh":"離散結構核對=([[0, 3], [0, 4], [1, 2], [2, 0], [2, 1]], [2, 1, 2], 否, 不成反函數)；先把每列(x,f(x))交換，再查每個目標是否恰有一個原像。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換關係與反函數門檻"]},"variant":{"family_id":"qf-density17-bij-swapped-relation-audit","seed":202610171050,"parameters":{"n":5,"m":3,"images":[2,2,1,0,0],"domain":"Fin_total_table","operation":"bij-swapped-relation-audit","semantic_template":"qf-density17-bij-swapped-relation-audit/{\"images\":[2,2,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0051","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(0)=∅→Fin(2)=[0, 1]；總函數f依輸入索引列images=[]。交換目標標籤a=0與b=1，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([], [], [[], []], 否, 否, 是)","explanation_zh":"離散結構核對=([], [], [[], []], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171051,"parameters":{"n":0,"m":2,"images":[],"a":0,"b":1,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[],\"images\":[],\"m\":2,\"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-20261017-r01-bijective-0052","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(2)=[0, 1]→Fin(2)=[0, 1]；總函數f依輸入索引列images=[0, 1]。交換目標標籤a=0與b=1，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([0, 1], [1, 0], [[1], [0]], 是, 是, 是)","explanation_zh":"離散結構核對=([0, 1], [1, 0], [[1], [0]], 是, 是, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171052,"parameters":{"n":2,"m":2,"images":[0,1],"a":0,"b":1,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[1,0],\"images\":[0,1],\"m\":2,\"n\":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-20261017-r01-bijective-0053","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(3)=[0, 1, 2]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 0, 1]。交換目標標籤a=0與b=1，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([0, 0, 1], [1, 1, 0], [[2], [0, 1], []], 否, 否, 是)","explanation_zh":"離散結構核對=([0, 0, 1], [1, 1, 0], [[2], [0, 1], []], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171053,"parameters":{"n":3,"m":3,"images":[0,0,1],"a":0,"b":1,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[1,1,0],\"images\":[0,0,1],\"m\":3,\"n\":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-20261017-r01-bijective-0054","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 0, 0, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([0, 0, 0, 0, 0], [2, 2, 2, 2, 2], [[], [], [0, 1, 2, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([0, 0, 0, 0, 0], [2, 2, 2, 2, 2], [[], [], [0, 1, 2, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171054,"parameters":{"n":5,"m":3,"images":[0,0,0,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[2,2,2,2,2],\"images\":[0,0,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0055","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 0, 0, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([1, 0, 0, 0, 0], [1, 2, 2, 2, 2], [[], [0], [1, 2, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([1, 0, 0, 0, 0], [1, 2, 2, 2, 2], [[], [0], [1, 2, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171055,"parameters":{"n":5,"m":3,"images":[1,0,0,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[1,2,2,2,2],\"images\":[1,0,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0056","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 0, 0, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([2, 0, 0, 0, 0], [0, 2, 2, 2, 2], [[0], [], [1, 2, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([2, 0, 0, 0, 0], [0, 2, 2, 2, 2], [[0], [], [1, 2, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171056,"parameters":{"n":5,"m":3,"images":[2,0,0,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[0,2,2,2,2],\"images\":[2,0,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0057","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1, 0, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([0, 1, 0, 0, 0], [2, 1, 2, 2, 2], [[], [1], [0, 2, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([0, 1, 0, 0, 0], [2, 1, 2, 2, 2], [[], [1], [0, 2, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171057,"parameters":{"n":5,"m":3,"images":[0,1,0,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[2,1,2,2,2],\"images\":[0,1,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0058","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 1, 0, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([1, 1, 0, 0, 0], [1, 1, 2, 2, 2], [[], [0, 1], [2, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([1, 1, 0, 0, 0], [1, 1, 2, 2, 2], [[], [0, 1], [2, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171058,"parameters":{"n":5,"m":3,"images":[1,1,0,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[1,1,2,2,2],\"images\":[1,1,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0059","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 1, 0, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], [0, 1, 2, 2, 2], [[0], [1], [2, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([2, 1, 0, 0, 0], [0, 1, 2, 2, 2], [[0], [1], [2, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171059,"parameters":{"n":5,"m":3,"images":[2,1,0,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[0,1,2,2,2],\"images\":[2,1,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0060","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 2, 0, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([0, 2, 0, 0, 0], [2, 0, 2, 2, 2], [[1], [], [0, 2, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([0, 2, 0, 0, 0], [2, 0, 2, 2, 2], [[1], [], [0, 2, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171060,"parameters":{"n":5,"m":3,"images":[0,2,0,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[2,0,2,2,2],\"images\":[0,2,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0061","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 2, 0, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([1, 2, 0, 0, 0], [1, 0, 2, 2, 2], [[1], [0], [2, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([1, 2, 0, 0, 0], [1, 0, 2, 2, 2], [[1], [0], [2, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171061,"parameters":{"n":5,"m":3,"images":[1,2,0,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[1,0,2,2,2],\"images\":[1,2,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0062","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 2, 0, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([2, 2, 0, 0, 0], [0, 0, 2, 2, 2], [[0, 1], [], [2, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([2, 2, 0, 0, 0], [0, 0, 2, 2, 2], [[0, 1], [], [2, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171062,"parameters":{"n":5,"m":3,"images":[2,2,0,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[0,0,2,2,2],\"images\":[2,2,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0063","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 0, 1, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([0, 0, 1, 0, 0], [2, 2, 1, 2, 2], [[], [2], [0, 1, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([0, 0, 1, 0, 0], [2, 2, 1, 2, 2], [[], [2], [0, 1, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171063,"parameters":{"n":5,"m":3,"images":[0,0,1,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[2,2,1,2,2],\"images\":[0,0,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0064","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 0, 1, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([1, 0, 1, 0, 0], [1, 2, 1, 2, 2], [[], [0, 2], [1, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([1, 0, 1, 0, 0], [1, 2, 1, 2, 2], [[], [0, 2], [1, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171064,"parameters":{"n":5,"m":3,"images":[1,0,1,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[1,2,1,2,2],\"images\":[1,0,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0065","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 0, 1, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([2, 0, 1, 0, 0], [0, 2, 1, 2, 2], [[0], [2], [1, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([2, 0, 1, 0, 0], [0, 2, 1, 2, 2], [[0], [2], [1, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171065,"parameters":{"n":5,"m":3,"images":[2,0,1,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[0,2,1,2,2],\"images\":[2,0,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0066","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1, 1, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([0, 1, 1, 0, 0], [2, 1, 1, 2, 2], [[], [1, 2], [0, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([0, 1, 1, 0, 0], [2, 1, 1, 2, 2], [[], [1, 2], [0, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171066,"parameters":{"n":5,"m":3,"images":[0,1,1,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[2,1,1,2,2],\"images\":[0,1,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0067","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 1, 1, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([1, 1, 1, 0, 0], [1, 1, 1, 2, 2], [[], [0, 1, 2], [3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([1, 1, 1, 0, 0], [1, 1, 1, 2, 2], [[], [0, 1, 2], [3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171067,"parameters":{"n":5,"m":3,"images":[1,1,1,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[1,1,1,2,2],\"images\":[1,1,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0068","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 1, 1, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([2, 1, 1, 0, 0], [0, 1, 1, 2, 2], [[0], [1, 2], [3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([2, 1, 1, 0, 0], [0, 1, 1, 2, 2], [[0], [1, 2], [3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171068,"parameters":{"n":5,"m":3,"images":[2,1,1,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[0,1,1,2,2],\"images\":[2,1,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0069","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 2, 1, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([0, 2, 1, 0, 0], [2, 0, 1, 2, 2], [[1], [2], [0, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([0, 2, 1, 0, 0], [2, 0, 1, 2, 2], [[1], [2], [0, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171069,"parameters":{"n":5,"m":3,"images":[0,2,1,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[2,0,1,2,2],\"images\":[0,2,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0070","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 2, 1, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([1, 2, 1, 0, 0], [1, 0, 1, 2, 2], [[1], [0, 2], [3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([1, 2, 1, 0, 0], [1, 0, 1, 2, 2], [[1], [0, 2], [3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171070,"parameters":{"n":5,"m":3,"images":[1,2,1,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[1,0,1,2,2],\"images\":[1,2,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0071","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 2, 1, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([2, 2, 1, 0, 0], [0, 0, 1, 2, 2], [[0, 1], [2], [3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([2, 2, 1, 0, 0], [0, 0, 1, 2, 2], [[0, 1], [2], [3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171071,"parameters":{"n":5,"m":3,"images":[2,2,1,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[0,0,1,2,2],\"images\":[2,2,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0072","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 0, 2, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([0, 0, 2, 0, 0], [2, 2, 0, 2, 2], [[2], [], [0, 1, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([0, 0, 2, 0, 0], [2, 2, 0, 2, 2], [[2], [], [0, 1, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171072,"parameters":{"n":5,"m":3,"images":[0,0,2,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[2,2,0,2,2],\"images\":[0,0,2,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0073","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 0, 2, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([1, 0, 2, 0, 0], [1, 2, 0, 2, 2], [[2], [0], [1, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([1, 0, 2, 0, 0], [1, 2, 0, 2, 2], [[2], [0], [1, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171073,"parameters":{"n":5,"m":3,"images":[1,0,2,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[1,2,0,2,2],\"images\":[1,0,2,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0074","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 0, 2, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([2, 0, 2, 0, 0], [0, 2, 0, 2, 2], [[0, 2], [], [1, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([2, 0, 2, 0, 0], [0, 2, 0, 2, 2], [[0, 2], [], [1, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171074,"parameters":{"n":5,"m":3,"images":[2,0,2,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[0,2,0,2,2],\"images\":[2,0,2,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0075","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"交換兩個目標標籤","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1, 2, 0, 0]。交換目標標籤a=0與b=2，其他目標不變。答 (原images, 交換後images, 新全目標纖維, 原雙射資格, 新雙射資格, 兩資格相同與否)。","answer":{"kind":"expression","value":"([0, 1, 2, 0, 0], [2, 1, 0, 2, 2], [[2], [1], [0, 3, 4]], 否, 否, 是)","explanation_zh":"離散結構核對=([0, 1, 2, 0, 0], [2, 1, 0, 2, 2], [[2], [1], [0, 3, 4]], 否, 否, 是)；只交換指定兩個輸出標籤，再對新表重新分纖維。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩個目標標籤"]},"variant":{"family_id":"qf-density17-bij-output-transposition","seed":202610171075,"parameters":{"n":5,"m":3,"images":[0,1,2,0,0],"a":0,"b":2,"domain":"Fin_total_table","operation":"bij-output-transposition","semantic_template":"qf-density17-bij-output-transposition/{\"after\":[2,1,0,2,2],\"images\":[0,1,2,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0076","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(0)=∅→Fin(0)=∅；總函數f依輸入索引列images=[]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([0, 0], [], 是, [], 是)","explanation_zh":"離散結構核對=([0, 0], [], 是, [], 是)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171076,"parameters":{"n":0,"m":0,"images":[],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[],\"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-20261017-r01-bijective-0077","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(0)=∅→Fin(2)=[0, 1]；總函數f依輸入索引列images=[]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([0, 2], [], 否, [[], []], 否)","explanation_zh":"離散結構核對=([0, 2], [], 否, [[], []], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171077,"parameters":{"n":0,"m":2,"images":[],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[],\"m\":2,\"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-20261017-r01-bijective-0078","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(1)=[0]→Fin(1)=[0]；總函數f依輸入索引列images=[0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([1, 1], [0], 是, [[0]], 是)","explanation_zh":"離散結構核對=([1, 1], [0], 是, [[0]], 是)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171078,"parameters":{"n":1,"m":1,"images":[0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[0],\"m\":1,\"n\":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-20261017-r01-bijective-0079","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(3)=[0, 1, 2]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1, 2]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([3, 3], [0, 1, 2], 是, [[0], [1], [2]], 是)","explanation_zh":"離散結構核對=([3, 3], [0, 1, 2], 是, [[0], [1], [2]], 是)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171079,"parameters":{"n":3,"m":3,"images":[0,1,2],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[0,1,2],\"m\":3,\"n\":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-20261017-r01-bijective-0080","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(3)=[0, 1, 2]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 0, 1]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([3, 3], [0, 0, 1], 是, [[0, 1], [2], []], 否)","explanation_zh":"離散結構核對=([3, 3], [0, 0, 1], 是, [[0, 1], [2], []], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171080,"parameters":{"n":3,"m":3,"images":[0,0,1],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[0,0,1],\"m\":3,\"n\":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-20261017-r01-bijective-0081","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(3)=[0, 1, 2]→Fin(2)=[0, 1]；總函數f依輸入索引列images=[0, 1, 1]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([3, 2], [0, 1, 1], 否, [[0], [1, 2]], 否)","explanation_zh":"離散結構核對=([3, 2], [0, 1, 1], 否, [[0], [1, 2]], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171081,"parameters":{"n":3,"m":2,"images":[0,1,1],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[0,1,1],\"m\":2,\"n\":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-20261017-r01-bijective-0082","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(2)=[0, 1]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([2, 3], [0, 1], 否, [[0], [1], []], 否)","explanation_zh":"離散結構核對=([2, 3], [0, 1], 否, [[0], [1], []], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171082,"parameters":{"n":2,"m":3,"images":[0,1],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[0,1],\"m\":3,\"n\":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-20261017-r01-bijective-0083","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 0, 0, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [0, 0, 0, 0, 0], 否, [[0, 1, 2, 3, 4], [], []], 否)","explanation_zh":"離散結構核對=([5, 3], [0, 0, 0, 0, 0], 否, [[0, 1, 2, 3, 4], [], []], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171083,"parameters":{"n":5,"m":3,"images":[0,0,0,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[0,0,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0084","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 0, 0, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [1, 0, 0, 0, 0], 否, [[1, 2, 3, 4], [0], []], 否)","explanation_zh":"離散結構核對=([5, 3], [1, 0, 0, 0, 0], 否, [[1, 2, 3, 4], [0], []], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171084,"parameters":{"n":5,"m":3,"images":[1,0,0,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[1,0,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0085","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 0, 0, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [2, 0, 0, 0, 0], 否, [[1, 2, 3, 4], [], [0]], 否)","explanation_zh":"離散結構核對=([5, 3], [2, 0, 0, 0, 0], 否, [[1, 2, 3, 4], [], [0]], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171085,"parameters":{"n":5,"m":3,"images":[2,0,0,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[2,0,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0086","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1, 0, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [0, 1, 0, 0, 0], 否, [[0, 2, 3, 4], [1], []], 否)","explanation_zh":"離散結構核對=([5, 3], [0, 1, 0, 0, 0], 否, [[0, 2, 3, 4], [1], []], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171086,"parameters":{"n":5,"m":3,"images":[0,1,0,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[0,1,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0087","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 1, 0, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [1, 1, 0, 0, 0], 否, [[2, 3, 4], [0, 1], []], 否)","explanation_zh":"離散結構核對=([5, 3], [1, 1, 0, 0, 0], 否, [[2, 3, 4], [0, 1], []], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171087,"parameters":{"n":5,"m":3,"images":[1,1,0,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[1,1,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0088","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 1, 0, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [2, 1, 0, 0, 0], 否, [[2, 3, 4], [1], [0]], 否)","explanation_zh":"離散結構核對=([5, 3], [2, 1, 0, 0, 0], 否, [[2, 3, 4], [1], [0]], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171088,"parameters":{"n":5,"m":3,"images":[2,1,0,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[2,1,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0089","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 2, 0, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [0, 2, 0, 0, 0], 否, [[0, 2, 3, 4], [], [1]], 否)","explanation_zh":"離散結構核對=([5, 3], [0, 2, 0, 0, 0], 否, [[0, 2, 3, 4], [], [1]], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171089,"parameters":{"n":5,"m":3,"images":[0,2,0,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[0,2,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0090","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 2, 0, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [1, 2, 0, 0, 0], 否, [[2, 3, 4], [0], [1]], 否)","explanation_zh":"離散結構核對=([5, 3], [1, 2, 0, 0, 0], 否, [[2, 3, 4], [0], [1]], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171090,"parameters":{"n":5,"m":3,"images":[1,2,0,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[1,2,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0091","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 2, 0, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [2, 2, 0, 0, 0], 否, [[2, 3, 4], [], [0, 1]], 否)","explanation_zh":"離散結構核對=([5, 3], [2, 2, 0, 0, 0], 否, [[2, 3, 4], [], [0, 1]], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171091,"parameters":{"n":5,"m":3,"images":[2,2,0,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[2,2,0,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0092","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 0, 1, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [0, 0, 1, 0, 0], 否, [[0, 1, 3, 4], [2], []], 否)","explanation_zh":"離散結構核對=([5, 3], [0, 0, 1, 0, 0], 否, [[0, 1, 3, 4], [2], []], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171092,"parameters":{"n":5,"m":3,"images":[0,0,1,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[0,0,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0093","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 0, 1, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [1, 0, 1, 0, 0], 否, [[1, 3, 4], [0, 2], []], 否)","explanation_zh":"離散結構核對=([5, 3], [1, 0, 1, 0, 0], 否, [[1, 3, 4], [0, 2], []], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171093,"parameters":{"n":5,"m":3,"images":[1,0,1,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[1,0,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0094","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 0, 1, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [2, 0, 1, 0, 0], 否, [[1, 3, 4], [2], [0]], 否)","explanation_zh":"離散結構核對=([5, 3], [2, 0, 1, 0, 0], 否, [[1, 3, 4], [2], [0]], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171094,"parameters":{"n":5,"m":3,"images":[2,0,1,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[2,0,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0095","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 1, 1, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [0, 1, 1, 0, 0], 否, [[0, 3, 4], [1, 2], []], 否)","explanation_zh":"離散結構核對=([5, 3], [0, 1, 1, 0, 0], 否, [[0, 3, 4], [1, 2], []], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171095,"parameters":{"n":5,"m":3,"images":[0,1,1,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[0,1,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0096","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 1, 1, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [1, 1, 1, 0, 0], 否, [[3, 4], [0, 1, 2], []], 否)","explanation_zh":"離散結構核對=([5, 3], [1, 1, 1, 0, 0], 否, [[3, 4], [0, 1, 2], []], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171096,"parameters":{"n":5,"m":3,"images":[1,1,1,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[1,1,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0097","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 1, 1, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [2, 1, 1, 0, 0], 否, [[3, 4], [1, 2], [0]], 否)","explanation_zh":"離散結構核對=([5, 3], [2, 1, 1, 0, 0], 否, [[3, 4], [1, 2], [0]], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171097,"parameters":{"n":5,"m":3,"images":[2,1,1,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[2,1,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0098","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[0, 2, 1, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [0, 2, 1, 0, 0], 否, [[0, 3, 4], [2], [1]], 否)","explanation_zh":"離散結構核對=([5, 3], [0, 2, 1, 0, 0], 否, [[0, 3, 4], [2], [1]], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171098,"parameters":{"n":5,"m":3,"images":[0,2,1,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[0,2,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0099","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[1, 2, 1, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [1, 2, 1, 0, 0], 否, [[3, 4], [0, 2], [1]], 否)","explanation_zh":"離散結構核對=([5, 3], [1, 2, 1, 0, 0], 否, [[3, 4], [0, 2], [1]], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171099,"parameters":{"n":5,"m":3,"images":[1,2,1,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[1,2,1,0,0],\"m\":3,\"n\":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-20261017-r01-bijective-0100","batch_id":"batch-20261012-advance-20261017-r01-bijective-foundations","language":"zh-Hant","source_mko_ids":["mko-bijective"],"title_zh":"存在性與給定表分開","stem_zh":"Fin型別從0起，完整總表。Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]；總函數f依輸入索引列images=[2, 2, 1, 0, 0]。答 ([n,m], images, 是否存在某個雙射, 給定表的全目標纖維, 給定f雙射資格)。存在性只問兩個有限型別能否配對。","answer":{"kind":"expression","value":"([5, 3], [2, 2, 1, 0, 0], 否, [[3, 4], [2], [0, 1]], 否)","explanation_zh":"離散結構核對=([5, 3], [2, 2, 1, 0, 0], 否, [[3, 4], [2], [0, 1]], 否)；大小決定某個雙射的存在可能；給定表另以每個纖維恰一元素檢查。；有限精確核對不是一般形式證明；基數相等不保證給定函數雙射；先資格後求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["存在性與給定表分開"]},"variant":{"family_id":"qf-density17-bij-size-obstruction-with-table","seed":202610171100,"parameters":{"n":5,"m":3,"images":[2,2,1,0,0],"domain":"Fin_total_table","operation":"bij-size-obstruction-with-table","semantic_template":"qf-density17-bij-size-obstruction-with-table/{\"images\":[2,2,1,0,0],\"m\":3,\"n\":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"}}