{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0001","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([], 是, [], [], [])","explanation_zh":"離散結構核對=([], 是, [], [], [])；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172001,"parameters":{"n":0,"m":0,"images":[],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0002","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172002,"parameters":{"n":0,"m":2,"images":[],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0003","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([0], 是, [0], [0], [0])","explanation_zh":"離散結構核對=([0], 是, [0], [0], [0])；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172003,"parameters":{"n":1,"m":1,"images":[0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0004","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([0, 1, 2], 是, [0, 1, 2], [0, 1, 2], [0, 1, 2])","explanation_zh":"離散結構核對=([0, 1, 2], 是, [0, 1, 2], [0, 1, 2], [0, 1, 2])；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172004,"parameters":{"n":3,"m":3,"images":[0,1,2],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0005","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([0, 0, 1], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([0, 0, 1], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172005,"parameters":{"n":3,"m":3,"images":[0,0,1],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0006","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([0, 1, 1], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([0, 1, 1], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172006,"parameters":{"n":3,"m":2,"images":[0,1,1],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0007","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([0, 1], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([0, 1], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172007,"parameters":{"n":2,"m":3,"images":[0,1],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0008","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([0, 0, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([0, 0, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172008,"parameters":{"n":5,"m":3,"images":[0,0,0,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0009","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([1, 0, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([1, 0, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172009,"parameters":{"n":5,"m":3,"images":[1,0,0,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0010","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([2, 0, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([2, 0, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172010,"parameters":{"n":5,"m":3,"images":[2,0,0,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0011","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([0, 1, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([0, 1, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172011,"parameters":{"n":5,"m":3,"images":[0,1,0,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0012","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([1, 1, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([1, 1, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172012,"parameters":{"n":5,"m":3,"images":[1,1,0,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0013","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([2, 1, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172013,"parameters":{"n":5,"m":3,"images":[2,1,0,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0014","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([0, 2, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([0, 2, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172014,"parameters":{"n":5,"m":3,"images":[0,2,0,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0015","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([1, 2, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([1, 2, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172015,"parameters":{"n":5,"m":3,"images":[1,2,0,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0016","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([2, 2, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([2, 2, 0, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172016,"parameters":{"n":5,"m":3,"images":[2,2,0,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0017","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([0, 0, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([0, 0, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172017,"parameters":{"n":5,"m":3,"images":[0,0,1,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0018","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([1, 0, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([1, 0, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172018,"parameters":{"n":5,"m":3,"images":[1,0,1,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0019","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([2, 0, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([2, 0, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172019,"parameters":{"n":5,"m":3,"images":[2,0,1,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0020","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([0, 1, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([0, 1, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172020,"parameters":{"n":5,"m":3,"images":[0,1,1,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0021","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([1, 1, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([1, 1, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172021,"parameters":{"n":5,"m":3,"images":[1,1,1,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0022","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([2, 1, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([2, 1, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172022,"parameters":{"n":5,"m":3,"images":[2,1,1,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0023","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([0, 2, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([0, 2, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172023,"parameters":{"n":5,"m":3,"images":[0,2,1,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0024","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([1, 2, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([1, 2, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172024,"parameters":{"n":5,"m":3,"images":[1,2,1,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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-r02-inverse-function-0025","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數總表與兩側還原","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 (images, f雙射資格, 反函數g總表或不成反函數, g(f(x))完整表或不求值, f(g(y))完整表或不求值)。","answer":{"kind":"expression","value":"([2, 2, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)","explanation_zh":"離散結構核對=([2, 2, 1, 0, 0], 否, 不成反函數, 不求值, 不求值)；只有每個目標恰有一原像才建立g，再分別算兩個合成。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數總表與兩側還原"]},"variant":{"family_id":"qf-density17-inverse-bijection-gate-and-table","seed":202610172025,"parameters":{"n":5,"m":3,"images":[2,2,1,0,0],"domain":"Fin_total_table","operation":"inverse-bijection-gate-and-table","semantic_template":"qf-density17-inverse-bijection-gate-and-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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0026","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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=[]。候選g的完整資料=[]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([], [], 是, [], [], 是, 是)","explanation_zh":"離散結構核對=([], [], 是, [], [], 是, 是)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172026,"parameters":{"n":0,"m":0,"images":[],"g":[],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[],\"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-r02-inverse-function-0027","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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=[]。候選g的完整資料=[-1, 0]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([], [否, 否], 否, 不求值, 不求值, 不求值, 不求值)","explanation_zh":"離散結構核對=([], [否, 否], 否, 不求值, 不求值, 不求值, 不求值)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172027,"parameters":{"n":0,"m":2,"images":[],"g":[-1,0],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[false,false],\"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-r02-inverse-function-0028","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 1, 0]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([0, 1], [是, 是, 是], 是, [0, 1], [0, 1, 0], 是, 否)","explanation_zh":"離散結構核對=([0, 1], [是, 是, 是], 是, [0, 1], [0, 1, 0], 是, 否)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172028,"parameters":{"n":2,"m":3,"images":[0,1],"g":[0,1,0],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[0,1,0],\"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-r02-inverse-function-0029","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 1]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([0, 1, 1], [是, 是], 是, [0, 1, 1], [0, 1], 否, 是)","explanation_zh":"離散結構核對=([0, 1, 1], [是, 是], 是, [0, 1, 1], [0, 1], 否, 是)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172029,"parameters":{"n":3,"m":2,"images":[0,1,1],"g":[0,1],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[0,1],\"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-r02-inverse-function-0030","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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=[1, 0]。候選g的完整資料=[1, 0]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([1, 0], [是, 是], 是, [0, 1], [0, 1], 是, 是)","explanation_zh":"離散結構核對=([1, 0], [是, 是], 是, [0, 1], [0, 1], 是, 是)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172030,"parameters":{"n":2,"m":2,"images":[1,0],"g":[1,0],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[1,0],\"images\":[1,0],\"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-r02-inverse-function-0031","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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=[1, 0]。候選g的完整資料=[0, 1]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([1, 0], [是, 是], 是, [1, 0], [1, 0], 否, 否)","explanation_zh":"離散結構核對=([1, 0], [是, 是], 是, [1, 0], [1, 0], 否, 否)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172031,"parameters":{"n":2,"m":2,"images":[1,0],"g":[0,1],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[0,1],\"images\":[1,0],\"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-r02-inverse-function-0032","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([0, 1], [是, 否], 否, 不求值, 不求值, 不求值, 不求值)","explanation_zh":"離散結構核對=([0, 1], [是, 否], 否, 不求值, 不求值, 不求值, 不求值)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172032,"parameters":{"n":2,"m":2,"images":[0,1],"g":[0,2],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[true,false],\"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-r02-inverse-function-0033","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 5]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([0, 0, 0, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)","explanation_zh":"離散結構核對=([0, 0, 0, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172033,"parameters":{"n":5,"m":3,"images":[0,0,0,0,0],"g":[0,2,5],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[true,true,false],\"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-r02-inverse-function-0034","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 4]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([1, 0, 0, 0, 0], [是, 是, 是], 是, [2, 0, 0, 0, 0], [1, 0, 0], 否, 否)","explanation_zh":"離散結構核對=([1, 0, 0, 0, 0], [是, 是, 是], 是, [2, 0, 0, 0, 0], [1, 0, 0], 否, 否)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172034,"parameters":{"n":5,"m":3,"images":[1,0,0,0,0],"g":[0,2,4],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[0,2,4],\"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-r02-inverse-function-0035","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 5]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([2, 0, 0, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)","explanation_zh":"離散結構核對=([2, 0, 0, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172035,"parameters":{"n":5,"m":3,"images":[2,0,0,0,0],"g":[0,2,5],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[true,true,false],\"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-r02-inverse-function-0036","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 4]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([0, 1, 0, 0, 0], [是, 是, 是], 是, [0, 2, 0, 0, 0], [0, 0, 0], 否, 否)","explanation_zh":"離散結構核對=([0, 1, 0, 0, 0], [是, 是, 是], 是, [0, 2, 0, 0, 0], [0, 0, 0], 否, 否)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172036,"parameters":{"n":5,"m":3,"images":[0,1,0,0,0],"g":[0,2,4],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[0,2,4],\"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-r02-inverse-function-0037","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 5]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([1, 1, 0, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)","explanation_zh":"離散結構核對=([1, 1, 0, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172037,"parameters":{"n":5,"m":3,"images":[1,1,0,0,0],"g":[0,2,5],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[true,true,false],\"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-r02-inverse-function-0038","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 4]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], [是, 是, 是], 是, [4, 2, 0, 0, 0], [2, 0, 0], 否, 否)","explanation_zh":"離散結構核對=([2, 1, 0, 0, 0], [是, 是, 是], 是, [4, 2, 0, 0, 0], [2, 0, 0], 否, 否)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172038,"parameters":{"n":5,"m":3,"images":[2,1,0,0,0],"g":[0,2,4],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[0,2,4],\"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-r02-inverse-function-0039","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 5]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([0, 2, 0, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)","explanation_zh":"離散結構核對=([0, 2, 0, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172039,"parameters":{"n":5,"m":3,"images":[0,2,0,0,0],"g":[0,2,5],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[true,true,false],\"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-r02-inverse-function-0040","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 4]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([1, 2, 0, 0, 0], [是, 是, 是], 是, [2, 4, 0, 0, 0], [1, 0, 0], 否, 否)","explanation_zh":"離散結構核對=([1, 2, 0, 0, 0], [是, 是, 是], 是, [2, 4, 0, 0, 0], [1, 0, 0], 否, 否)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172040,"parameters":{"n":5,"m":3,"images":[1,2,0,0,0],"g":[0,2,4],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[0,2,4],\"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-r02-inverse-function-0041","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 5]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([2, 2, 0, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)","explanation_zh":"離散結構核對=([2, 2, 0, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172041,"parameters":{"n":5,"m":3,"images":[2,2,0,0,0],"g":[0,2,5],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[true,true,false],\"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-r02-inverse-function-0042","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 4]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([0, 0, 1, 0, 0], [是, 是, 是], 是, [0, 0, 2, 0, 0], [0, 1, 0], 否, 否)","explanation_zh":"離散結構核對=([0, 0, 1, 0, 0], [是, 是, 是], 是, [0, 0, 2, 0, 0], [0, 1, 0], 否, 否)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172042,"parameters":{"n":5,"m":3,"images":[0,0,1,0,0],"g":[0,2,4],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[0,2,4],\"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-r02-inverse-function-0043","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 5]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([1, 0, 1, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)","explanation_zh":"離散結構核對=([1, 0, 1, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172043,"parameters":{"n":5,"m":3,"images":[1,0,1,0,0],"g":[0,2,5],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[true,true,false],\"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-r02-inverse-function-0044","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 4]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([2, 0, 1, 0, 0], [是, 是, 是], 是, [4, 0, 2, 0, 0], [2, 1, 0], 否, 否)","explanation_zh":"離散結構核對=([2, 0, 1, 0, 0], [是, 是, 是], 是, [4, 0, 2, 0, 0], [2, 1, 0], 否, 否)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172044,"parameters":{"n":5,"m":3,"images":[2,0,1,0,0],"g":[0,2,4],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[0,2,4],\"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-r02-inverse-function-0045","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 5]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([0, 1, 1, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)","explanation_zh":"離散結構核對=([0, 1, 1, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172045,"parameters":{"n":5,"m":3,"images":[0,1,1,0,0],"g":[0,2,5],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[true,true,false],\"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-r02-inverse-function-0046","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 4]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([1, 1, 1, 0, 0], [是, 是, 是], 是, [2, 2, 2, 0, 0], [1, 1, 0], 否, 否)","explanation_zh":"離散結構核對=([1, 1, 1, 0, 0], [是, 是, 是], 是, [2, 2, 2, 0, 0], [1, 1, 0], 否, 否)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172046,"parameters":{"n":5,"m":3,"images":[1,1,1,0,0],"g":[0,2,4],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[0,2,4],\"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-r02-inverse-function-0047","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 5]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([2, 1, 1, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)","explanation_zh":"離散結構核對=([2, 1, 1, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172047,"parameters":{"n":5,"m":3,"images":[2,1,1,0,0],"g":[0,2,5],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[true,true,false],\"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-r02-inverse-function-0048","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 4]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([0, 2, 1, 0, 0], [是, 是, 是], 是, [0, 4, 2, 0, 0], [0, 1, 0], 否, 否)","explanation_zh":"離散結構核對=([0, 2, 1, 0, 0], [是, 是, 是], 是, [0, 4, 2, 0, 0], [0, 1, 0], 否, 否)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172048,"parameters":{"n":5,"m":3,"images":[0,2,1,0,0],"g":[0,2,4],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[0,2,4],\"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-r02-inverse-function-0049","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 5]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([1, 2, 1, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)","explanation_zh":"離散結構核對=([1, 2, 1, 0, 0], [是, 是, 否], 否, 不求值, 不求值, 不求值, 不求值)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172049,"parameters":{"n":5,"m":3,"images":[1,2,1,0,0],"g":[0,2,5],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[true,true,false],\"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-r02-inverse-function-0050","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選逆表的兩側分開核對","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]。候選g的完整資料=[0, 2, 4]，其值可能不合Fin(n)。答 (images, g每格合型mask, g總表資格, g∘f表, f∘g表, 左identity資格, 右identity資格)。任一g格不合型，後四欄都答不求值；合型時單側identity不叫雙側反函數。","answer":{"kind":"expression","value":"([2, 2, 1, 0, 0], [是, 是, 是], 是, [4, 4, 2, 0, 0], [2, 1, 0], 否, 否)","explanation_zh":"離散結構核對=([2, 2, 1, 0, 0], [是, 是, 是], 是, [4, 4, 2, 0, 0], [2, 1, 0], 否, 否)；先驗g每個值；全合型才沿兩個方向查表並逐索引比identity。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選逆表的兩側分開核對"]},"variant":{"family_id":"qf-density17-inverse-candidate-two-sided","seed":202610172050,"parameters":{"n":5,"m":3,"images":[2,2,1,0,0],"g":[0,2,4],"domain":"Fin_total_table","operation":"inverse-candidate-two-sided","semantic_template":"qf-density17-inverse-candidate-two-sided/{\"g\":[0,2,4],\"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-r02-inverse-function-0051","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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=[]。有序兩查詢=[0, -1]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([], 是, [否, 否], [不合陪域, 不合陪域])","explanation_zh":"離散結構核對=([], 是, [否, 否], [不合陪域, 不合陪域])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172051,"parameters":{"n":0,"m":0,"images":[],"queries":[0,-1],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[],\"m\":0,\"n\":0,\"queries\":[\"untyped\",\"untyped\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0052","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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=[]。有序兩查詢=[0, 1]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172052,"parameters":{"n":0,"m":2,"images":[],"queries":[0,1],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[],\"m\":2,\"n\":0,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0053","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 1]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([0], 是, [是, 否], [0, 不合陪域])","explanation_zh":"離散結構核對=([0], 是, [是, 否], [0, 不合陪域])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172053,"parameters":{"n":1,"m":1,"images":[0],"queries":[0,1],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[0],\"m\":1,\"n\":1,\"queries\":[0,\"untyped\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0054","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[2, 1]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([0, 1, 2], 是, [是, 是], [2, 1])","explanation_zh":"離散結構核對=([0, 1, 2], 是, [是, 是], [2, 1])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172054,"parameters":{"n":3,"m":3,"images":[0,1,2],"queries":[2,1],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[0,1,2],\"m\":3,\"n\":3,\"queries\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0055","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[2, -1]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([0, 0, 1], 否, [是, 否], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([0, 0, 1], 否, [是, 否], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172055,"parameters":{"n":3,"m":3,"images":[0,0,1],"queries":[2,-1],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[0,0,1],\"m\":3,\"n\":3,\"queries\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0056","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([0, 1, 1], 否, [是, 否], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([0, 1, 1], 否, [是, 否], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172056,"parameters":{"n":3,"m":2,"images":[0,1,1],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[0,1,1],\"m\":2,\"n\":3,\"queries\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0057","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([0, 1], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([0, 1], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172057,"parameters":{"n":2,"m":3,"images":[0,1],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[0,1],\"m\":3,\"n\":2,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0058","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([0, 0, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([0, 0, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172058,"parameters":{"n":5,"m":3,"images":[0,0,0,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[0,0,0,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0059","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([1, 0, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([1, 0, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172059,"parameters":{"n":5,"m":3,"images":[1,0,0,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[1,0,0,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0060","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([2, 0, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([2, 0, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172060,"parameters":{"n":5,"m":3,"images":[2,0,0,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[2,0,0,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0061","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([0, 1, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([0, 1, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172061,"parameters":{"n":5,"m":3,"images":[0,1,0,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[0,1,0,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0062","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([1, 1, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([1, 1, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172062,"parameters":{"n":5,"m":3,"images":[1,1,0,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[1,1,0,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0063","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([2, 1, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172063,"parameters":{"n":5,"m":3,"images":[2,1,0,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[2,1,0,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0064","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([0, 2, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([0, 2, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172064,"parameters":{"n":5,"m":3,"images":[0,2,0,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[0,2,0,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0065","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([1, 2, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([1, 2, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172065,"parameters":{"n":5,"m":3,"images":[1,2,0,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[1,2,0,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0066","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([2, 2, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([2, 2, 0, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172066,"parameters":{"n":5,"m":3,"images":[2,2,0,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[2,2,0,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0067","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([0, 0, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([0, 0, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172067,"parameters":{"n":5,"m":3,"images":[0,0,1,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[0,0,1,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0068","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([1, 0, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([1, 0, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172068,"parameters":{"n":5,"m":3,"images":[1,0,1,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[1,0,1,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0069","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([2, 0, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([2, 0, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172069,"parameters":{"n":5,"m":3,"images":[2,0,1,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[2,0,1,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0070","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([0, 1, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([0, 1, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172070,"parameters":{"n":5,"m":3,"images":[0,1,1,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[0,1,1,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0071","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([1, 1, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([1, 1, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172071,"parameters":{"n":5,"m":3,"images":[1,1,1,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[1,1,1,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0072","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([2, 1, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([2, 1, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172072,"parameters":{"n":5,"m":3,"images":[2,1,1,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[2,1,1,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0073","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([0, 2, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([0, 2, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172073,"parameters":{"n":5,"m":3,"images":[0,2,1,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[0,2,1,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0074","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([1, 2, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([1, 2, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172074,"parameters":{"n":5,"m":3,"images":[1,2,1,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[1,2,1,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0075","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數查詢的陪域資格","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]。有序兩查詢=[0, 2]。答 (images, f雙射資格, 兩查詢在原Fin(m)中的資格, 兩結果)。先判f；非雙射每結果均不成反函數，雙射但查詢域外答不合陪域。","answer":{"kind":"expression","value":"([2, 2, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])","explanation_zh":"離散結構核對=([2, 2, 1, 0, 0], 否, [是, 是], [不成反函數, 不成反函數])；查詢資格按原宣告陪域；反函數未成立時不讀交換資料。；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；反函數不是函數值倒數；兩側identity分開。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數查詢的陪域資格"]},"variant":{"family_id":"qf-density17-inverse-query-qualification","seed":202610172075,"parameters":{"n":5,"m":3,"images":[2,2,1,0,0],"queries":[0,2],"domain":"Fin_total_table","operation":"inverse-query-qualification","semantic_template":"qf-density17-inverse-query-qualification/{\"images\":[2,2,1,0,0],\"m\":3,\"n\":5,\"queries\":[true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"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-r02-inverse-function-0076","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=-2、b=1，有序兩目標y=[(1)/(-2), (-3)/(2)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([-1/2, 1/2], [3/4, 5/4], [-1/2, -3/2], 是)","explanation_zh":"離散結構核對=([-1/2, 1/2], [3/4, 5/4], [-1/2, -3/2], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172076,"parameters":{"a":-2,"b":1,"ys":[[1,-2],[-3,2]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":-2,\"b\":1,\"ys\":[\"-1/2\",\"-3/2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0077","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=-2、b=-11，有序兩目標y=[(-11)/(3), (2)/(2)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([-1/2, -11/2], [-11/3, -6/1], [-11/3, 1/1], 是)","explanation_zh":"離散結構核對=([-1/2, -11/2], [-11/3, -6/1], [-11/3, 1/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172077,"parameters":{"a":-2,"b":-11,"ys":[[-11,3],[2,2]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":-2,\"b\":-11,\"ys\":[\"-11/3\",\"1/1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0078","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=3、b=-10，有序兩目標y=[(-10)/(3), (3)/(3)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([1/3, 10/3], [20/9, 11/3], [-10/3, 1/1], 是)","explanation_zh":"離散結構核對=([1/3, 10/3], [20/9, 11/3], [-10/3, 1/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172078,"parameters":{"a":3,"b":-10,"ys":[[-10,3],[3,3]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":3,\"b\":-10,\"ys\":[\"-10/3\",\"1/1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0079","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=-4、b=-9，有序兩目標y=[(-9)/(3), (4)/(4)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([-1/4, -9/4], [-3/2, -5/2], [-3/1, 1/1], 是)","explanation_zh":"離散結構核對=([-1/4, -9/4], [-3/2, -5/2], [-3/1, 1/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172079,"parameters":{"a":-4,"b":-9,"ys":[[-9,3],[4,4]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":-4,\"b\":-9,\"ys\":[\"-3/1\",\"1/1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0080","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=5、b=-8，有序兩目標y=[(-8)/(3), (5)/(1)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([1/5, 8/5], [16/15, 13/5], [-8/3, 5/1], 是)","explanation_zh":"離散結構核對=([1/5, 8/5], [16/15, 13/5], [-8/3, 5/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172080,"parameters":{"a":5,"b":-8,"ys":[[-8,3],[5,1]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":5,\"b\":-8,\"ys\":[\"-8/3\",\"5/1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0081","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=-1、b=-7，有序兩目標y=[(-7)/(3), (6)/(2)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([-1/1, -7/1], [-14/3, -10/1], [-7/3, 3/1], 是)","explanation_zh":"離散結構核對=([-1/1, -7/1], [-14/3, -10/1], [-7/3, 3/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172081,"parameters":{"a":-1,"b":-7,"ys":[[-7,3],[6,2]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":-1,\"b\":-7,\"ys\":[\"-7/3\",\"3/1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0082","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=2、b=-6，有序兩目標y=[(-6)/(3), (7)/(3)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([1/2, 3/1], [2/1, 25/6], [-2/1, 7/3], 是)","explanation_zh":"離散結構核對=([1/2, 3/1], [2/1, 25/6], [-2/1, 7/3], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172082,"parameters":{"a":2,"b":-6,"ys":[[-6,3],[7,3]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":2,\"b\":-6,\"ys\":[\"-2/1\",\"7/3\"]}"}},"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-r02-inverse-function-0083","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=-3、b=-5，有序兩目標y=[(-5)/(3), (8)/(4)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([-1/3, -5/3], [-10/9, -7/3], [-5/3, 2/1], 是)","explanation_zh":"離散結構核對=([-1/3, -5/3], [-10/9, -7/3], [-5/3, 2/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172083,"parameters":{"a":-3,"b":-5,"ys":[[-5,3],[8,4]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":-3,\"b\":-5,\"ys\":[\"-5/3\",\"2/1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0084","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=4、b=-4，有序兩目標y=[(-4)/(3), (9)/(1)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([1/4, 1/1], [2/3, 13/4], [-4/3, 9/1], 是)","explanation_zh":"離散結構核對=([1/4, 1/1], [2/3, 13/4], [-4/3, 9/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172084,"parameters":{"a":4,"b":-4,"ys":[[-4,3],[9,1]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":4,\"b\":-4,\"ys\":[\"-4/3\",\"9/1\"]}"}},"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-r02-inverse-function-0085","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=-5、b=-3，有序兩目標y=[(-3)/(3), (10)/(2)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([-1/5, -3/5], [-2/5, -8/5], [-1/1, 5/1], 是)","explanation_zh":"離散結構核對=([-1/5, -3/5], [-2/5, -8/5], [-1/1, 5/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172085,"parameters":{"a":-5,"b":-3,"ys":[[-3,3],[10,2]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":-5,\"b\":-3,\"ys\":[\"-1/1\",\"5/1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0086","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=1、b=-2，有序兩目標y=[(-2)/(3), (11)/(3)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([1/1, 2/1], [4/3, 17/3], [-2/3, 11/3], 是)","explanation_zh":"離散結構核對=([1/1, 2/1], [4/3, 17/3], [-2/3, 11/3], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172086,"parameters":{"a":1,"b":-2,"ys":[[-2,3],[11,3]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":1,\"b\":-2,\"ys\":[\"-2/3\",\"11/3\"]}"}},"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-r02-inverse-function-0087","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=-2、b=-1，有序兩目標y=[(-1)/(3), (12)/(4)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([-1/2, -1/2], [-1/3, -2/1], [-1/3, 3/1], 是)","explanation_zh":"離散結構核對=([-1/2, -1/2], [-1/3, -2/1], [-1/3, 3/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172087,"parameters":{"a":-2,"b":-1,"ys":[[-1,3],[12,4]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":-2,\"b\":-1,\"ys\":[\"-1/3\",\"3/1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0088","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=3、b=0，有序兩目標y=[(0)/(3), (13)/(1)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([1/3, 0/1], [0/1, 13/3], [0/1, 13/1], 是)","explanation_zh":"離散結構核對=([1/3, 0/1], [0/1, 13/3], [0/1, 13/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172088,"parameters":{"a":3,"b":0,"ys":[[0,3],[13,1]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":3,\"b\":0,\"ys\":[\"0/1\",\"13/1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0089","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=-4、b=1，有序兩目標y=[(1)/(3), (14)/(2)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([-1/4, 1/4], [1/6, -3/2], [1/3, 7/1], 是)","explanation_zh":"離散結構核對=([-1/4, 1/4], [1/6, -3/2], [1/3, 7/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172089,"parameters":{"a":-4,"b":1,"ys":[[1,3],[14,2]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":-4,\"b\":1,\"ys\":[\"1/3\",\"7/1\"]}"}},"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-r02-inverse-function-0090","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=5、b=2，有序兩目標y=[(2)/(3), (15)/(3)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([1/5, -2/5], [-4/15, 3/5], [2/3, 5/1], 是)","explanation_zh":"離散結構核對=([1/5, -2/5], [-4/15, 3/5], [2/3, 5/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172090,"parameters":{"a":5,"b":2,"ys":[[2,3],[15,3]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":5,\"b\":2,\"ys\":[\"2/3\",\"5/1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0091","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=-1、b=3，有序兩目標y=[(3)/(3), (16)/(4)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([-1/1, 3/1], [2/1, -1/1], [1/1, 4/1], 是)","explanation_zh":"離散結構核對=([-1/1, 3/1], [2/1, -1/1], [1/1, 4/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172091,"parameters":{"a":-1,"b":3,"ys":[[3,3],[16,4]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":-1,\"b\":3,\"ys\":[\"1/1\",\"4/1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0092","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=2、b=4，有序兩目標y=[(4)/(3), (17)/(1)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([1/2, -2/1], [-4/3, 13/2], [4/3, 17/1], 是)","explanation_zh":"離散結構核對=([1/2, -2/1], [-4/3, 13/2], [4/3, 17/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172092,"parameters":{"a":2,"b":4,"ys":[[4,3],[17,1]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":2,\"b\":4,\"ys\":[\"4/3\",\"17/1\"]}"}},"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-r02-inverse-function-0093","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=-3、b=5，有序兩目標y=[(5)/(3), (18)/(2)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([-1/3, 5/3], [10/9, -4/3], [5/3, 9/1], 是)","explanation_zh":"離散結構核對=([-1/3, 5/3], [10/9, -4/3], [5/3, 9/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172093,"parameters":{"a":-3,"b":5,"ys":[[5,3],[18,2]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":-3,\"b\":5,\"ys\":[\"5/3\",\"9/1\"]}"}},"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-r02-inverse-function-0094","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=4、b=6，有序兩目標y=[(6)/(3), (19)/(3)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([1/4, -3/2], [-1/1, 1/12], [2/1, 19/3], 是)","explanation_zh":"離散結構核對=([1/4, -3/2], [-1/1, 1/12], [2/1, 19/3], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172094,"parameters":{"a":4,"b":6,"ys":[[6,3],[19,3]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":4,\"b\":6,\"ys\":[\"2/1\",\"19/3\"]}"}},"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-r02-inverse-function-0095","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=-5、b=7，有序兩目標y=[(7)/(3), (20)/(4)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([-1/5, 7/5], [14/15, 2/5], [7/3, 5/1], 是)","explanation_zh":"離散結構核對=([-1/5, 7/5], [14/15, 2/5], [7/3, 5/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172095,"parameters":{"a":-5,"b":7,"ys":[[7,3],[20,4]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":-5,\"b\":7,\"ys\":[\"7/3\",\"5/1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r02-inverse-function-0096","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=1、b=8，有序兩目標y=[(8)/(3), (21)/(1)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([1/1, -8/1], [-16/3, 13/1], [8/3, 21/1], 是)","explanation_zh":"離散結構核對=([1/1, -8/1], [-16/3, 13/1], [8/3, 21/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172096,"parameters":{"a":1,"b":8,"ys":[[8,3],[21,1]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":1,\"b\":8,\"ys\":[\"8/3\",\"21/1\"]}"}},"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-r02-inverse-function-0097","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=-2、b=9，有序兩目標y=[(9)/(3), (22)/(2)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([-1/2, 9/2], [3/1, -1/1], [3/1, 11/1], 是)","explanation_zh":"離散結構核對=([-1/2, 9/2], [3/1, -1/1], [3/1, 11/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172097,"parameters":{"a":-2,"b":9,"ys":[[9,3],[22,2]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":-2,\"b\":9,\"ys\":[\"3/1\",\"11/1\"]}"}},"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-r02-inverse-function-0098","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=3、b=10，有序兩目標y=[(10)/(3), (23)/(3)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([1/3, -10/3], [-20/9, -7/9], [10/3, 23/3], 是)","explanation_zh":"離散結構核對=([1/3, -10/3], [-20/9, -7/9], [10/3, 23/3], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172098,"parameters":{"a":3,"b":10,"ys":[[10,3],[23,3]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":3,\"b\":10,\"ys\":[\"10/3\",\"23/3\"]}"}},"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-r02-inverse-function-0099","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=-4、b=11，有序兩目標y=[(11)/(3), (24)/(4)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([-1/4, 11/4], [11/6, 5/4], [11/3, 6/1], 是)","explanation_zh":"離散結構核對=([-1/4, 11/4], [11/6, 5/4], [11/3, 6/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172099,"parameters":{"a":-4,"b":11,"ys":[[11,3],[24,4]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":-4,\"b\":11,\"ys\":[\"11/3\",\"6/1\"]}"}},"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-r02-inverse-function-0100","batch_id":"batch-20261012-advance-20261017-r02-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率的精確兩點還原","stem_zh":"R到R仿射模型，斜率非零。f(x)=a*x+b，a=5、b=12，有序兩目標y=[(12)/(3), (25)/(1)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。","answer":{"kind":"expression","value":"([1/5, -12/5], [-8/5, 13/5], [4/1, 25/1], 是)","explanation_zh":"離散結構核對=([1/5, -12/5], [-8/5, 13/5], [4/1, 25/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率的精確兩點還原"]},"variant":{"family_id":"qf-density17-inverse-affine-recovery","seed":202610172100,"parameters":{"a":5,"b":12,"ys":[[12,3],[25,1]],"domain":"R_affine_nonzero_integer_slice","operation":"inverse-affine-recovery","semantic_template":"qf-density17-inverse-affine-recovery/{\"a\":5,\"b\":12,\"ys\":[\"4/1\",\"25/1\"]}"}},"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"}}