{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0001","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V=∅、E=∅，邊無方向、沒有自環或重邊。候選端點集合 {1, 0} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{1, 0}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044001,"parameters":{"V":[],"E":[],"operation":"edge_membership","u":1,"v":0,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[],\"V\":[],\"operation\":\"edge_membership\",\"u\":1,\"v\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0002","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={-11, -10, -9, -8}、E={{-11, -10}, {-10, -9}, {-9, -8}}，邊無方向、沒有自環或重邊。候選端點集合 {-11, -11} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{-11, -11}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044002,"parameters":{"V":[-11,-10,-9,-8],"E":[[-11,-10],[-10,-9],[-9,-8]],"operation":"edge_membership","u":-11,"v":-11,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-11,-10],[-10,-9],[-9,-8]],\"V\":[-11,-10,-9,-8],\"operation\":\"edge_membership\",\"u\":-11,\"v\":-11}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0003","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={-10, -9, -8, -7, -6}、E={{-10, -9}, {-10, -8}, {-10, -7}}，邊無方向、沒有自環或重邊。候選端點集合 {-10, -5} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{-10, -5}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044003,"parameters":{"V":[-10,-9,-8,-7,-6],"E":[[-10,-9],[-10,-8],[-10,-7]],"operation":"edge_membership","u":-10,"v":-5,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-10,-9],[-10,-8],[-10,-7]],\"V\":[-10,-9,-8,-7,-6],\"operation\":\"edge_membership\",\"u\":-10,\"v\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0004","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={-9, -8, -7, -6, -5, -4}、E={{-9, -8}, {-9, -7}, {-8, -7}}，邊無方向、沒有自環或重邊。候選端點集合 {-8, -9} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"是","explanation_zh":"{-8, -9}∈E=是。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044004,"parameters":{"V":[-9,-8,-7,-6,-5,-4],"E":[[-9,-8],[-9,-7],[-8,-7]],"operation":"edge_membership","u":-8,"v":-9,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-9,-8],[-9,-7],[-8,-7]],\"V\":[-9,-8,-7,-6,-5,-4],\"operation\":\"edge_membership\",\"u\":-8,\"v\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0005","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={-8, -7, -6}、E={{-8, -7}}，邊無方向、沒有自環或重邊。候選端點集合 {-8, -6} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{-8, -6}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044005,"parameters":{"V":[-8,-7,-6],"E":[[-8,-7]],"operation":"edge_membership","u":-8,"v":-6,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-8,-7]],\"V\":[-8,-7,-6],\"operation\":\"edge_membership\",\"u\":-8,\"v\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0006","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={-7, -6, -5, -4}、E={{-7, -6}, {-7, -5}, {-7, -4}, {-6, -5}, {-6, -4}, {-5, -4}}，邊無方向、沒有自環或重邊。候選端點集合 {-7, -3} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{-7, -3}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044006,"parameters":{"V":[-7,-6,-5,-4],"E":[[-7,-6],[-7,-5],[-7,-4],[-6,-5],[-6,-4],[-5,-4]],"operation":"edge_membership","u":-7,"v":-3,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-7,-6],[-7,-5],[-7,-4],[-6,-5],[-6,-4],[-5,-4]],\"V\":[-7,-6,-5,-4],\"operation\":\"edge_membership\",\"u\":-7,\"v\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0007","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={-6, -5, -4, -3, -2}、E=∅，邊無方向、沒有自環或重邊。候選端點集合 {-2, -6} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{-2, -6}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044007,"parameters":{"V":[-6,-5,-4,-3,-2],"E":[],"operation":"edge_membership","u":-2,"v":-6,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[],\"V\":[-6,-5,-4,-3,-2],\"operation\":\"edge_membership\",\"u\":-2,\"v\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0008","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={-5, -4, -3, -2, -1, 0}、E={{-5, -4}, {-4, -3}, {-3, -2}, {-2, -1}, {-1, 0}}，邊無方向、沒有自環或重邊。候選端點集合 {-5, 0} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{-5, 0}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044008,"parameters":{"V":[-5,-4,-3,-2,-1,0],"E":[[-5,-4],[-4,-3],[-3,-2],[-2,-1],[-1,0]],"operation":"edge_membership","u":-5,"v":0,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-5,-4],[-4,-3],[-3,-2],[-2,-1],[-1,0]],\"V\":[-5,-4,-3,-2,-1,0],\"operation\":\"edge_membership\",\"u\":-5,\"v\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0009","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={-4, -3, -2}、E={{-4, -3}}，邊無方向、沒有自環或重邊。候選端點集合 {-4, -1} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{-4, -1}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044009,"parameters":{"V":[-4,-3,-2],"E":[[-4,-3]],"operation":"edge_membership","u":-4,"v":-1,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-4,-3]],\"V\":[-4,-3,-2],\"operation\":\"edge_membership\",\"u\":-4,\"v\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0010","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={-3, -2, -1, 0}、E={{-3, -2}, {-3, -1}, {-2, -1}}，邊無方向、沒有自環或重邊。候選端點集合 {-2, -3} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"是","explanation_zh":"{-2, -3}∈E=是。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044010,"parameters":{"V":[-3,-2,-1,0],"E":[[-3,-2],[-3,-1],[-2,-1]],"operation":"edge_membership","u":-2,"v":-3,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-3,-2],[-3,-1],[-2,-1]],\"V\":[-3,-2,-1,0],\"operation\":\"edge_membership\",\"u\":-2,\"v\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0011","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={-2, -1, 0, 1, 2}、E={{-2, -1}, {0, 1}}，邊無方向、沒有自環或重邊。候選端點集合 {-2, 2} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{-2, 2}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044011,"parameters":{"V":[-2,-1,0,1,2],"E":[[-2,-1],[0,1]],"operation":"edge_membership","u":-2,"v":2,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-2,-1],[0,1]],\"V\":[-2,-1,0,1,2],\"operation\":\"edge_membership\",\"u\":-2,\"v\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0012","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={-1, 0, 1, 2, 3, 4}、E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}，邊無方向、沒有自環或重邊。候選端點集合 {-1, 5} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{-1, 5}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044012,"parameters":{"V":[-1,0,1,2,3,4],"E":[[-1,0],[-1,1],[-1,2],[-1,3],[-1,4],[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],"operation":"edge_membership","u":-1,"v":5,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-1,0],[-1,1],[-1,2],[-1,3],[-1,4],[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],\"V\":[-1,0,1,2,3,4],\"operation\":\"edge_membership\",\"u\":-1,\"v\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0013","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={0, 1, 2}、E=∅，邊無方向、沒有自環或重邊。候選端點集合 {2, 0} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{2, 0}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044013,"parameters":{"V":[0,1,2],"E":[],"operation":"edge_membership","u":2,"v":0,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[],\"V\":[0,1,2],\"operation\":\"edge_membership\",\"u\":2,\"v\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0014","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={1, 2, 3, 4}、E={{1, 2}, {2, 3}, {3, 4}}，邊無方向、沒有自環或重邊。候選端點集合 {1, 4} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{1, 4}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044014,"parameters":{"V":[1,2,3,4],"E":[[1,2],[2,3],[3,4]],"operation":"edge_membership","u":1,"v":4,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[1,2],[2,3],[3,4]],\"V\":[1,2,3,4],\"operation\":\"edge_membership\",\"u\":1,\"v\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0015","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={2, 3, 4, 5, 6}、E={{2, 3}, {2, 4}, {2, 5}}，邊無方向、沒有自環或重邊。候選端點集合 {2, 7} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{2, 7}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044015,"parameters":{"V":[2,3,4,5,6],"E":[[2,3],[2,4],[2,5]],"operation":"edge_membership","u":2,"v":7,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[2,3],[2,4],[2,5]],\"V\":[2,3,4,5,6],\"operation\":\"edge_membership\",\"u\":2,\"v\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0016","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={3, 4, 5, 6, 7, 8}、E={{3, 4}, {3, 5}, {4, 5}}，邊無方向、沒有自環或重邊。候選端點集合 {4, 3} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"是","explanation_zh":"{4, 3}∈E=是。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044016,"parameters":{"V":[3,4,5,6,7,8],"E":[[3,4],[3,5],[4,5]],"operation":"edge_membership","u":4,"v":3,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[3,4],[3,5],[4,5]],\"V\":[3,4,5,6,7,8],\"operation\":\"edge_membership\",\"u\":4,\"v\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0017","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={4, 5, 6}、E={{4, 5}}，邊無方向、沒有自環或重邊。候選端點集合 {4, 6} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{4, 6}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044017,"parameters":{"V":[4,5,6],"E":[[4,5]],"operation":"edge_membership","u":4,"v":6,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[4,5]],\"V\":[4,5,6],\"operation\":\"edge_membership\",\"u\":4,\"v\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0018","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={5, 6, 7, 8}、E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}，邊無方向、沒有自環或重邊。候選端點集合 {5, 9} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{5, 9}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044018,"parameters":{"V":[5,6,7,8],"E":[[5,6],[5,7],[5,8],[6,7],[6,8],[7,8]],"operation":"edge_membership","u":5,"v":9,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[5,6],[5,7],[5,8],[6,7],[6,8],[7,8]],\"V\":[5,6,7,8],\"operation\":\"edge_membership\",\"u\":5,\"v\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0019","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={6, 7, 8, 9, 10}、E=∅，邊無方向、沒有自環或重邊。候選端點集合 {10, 6} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{10, 6}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044019,"parameters":{"V":[6,7,8,9,10],"E":[],"operation":"edge_membership","u":10,"v":6,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[],\"V\":[6,7,8,9,10],\"operation\":\"edge_membership\",\"u\":10,\"v\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0020","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={7, 8, 9, 10, 11, 12}、E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}，邊無方向、沒有自環或重邊。候選端點集合 {7, 12} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{7, 12}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044020,"parameters":{"V":[7,8,9,10,11,12],"E":[[7,8],[8,9],[9,10],[10,11],[11,12]],"operation":"edge_membership","u":7,"v":12,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[7,8],[8,9],[9,10],[10,11],[11,12]],\"V\":[7,8,9,10,11,12],\"operation\":\"edge_membership\",\"u\":7,\"v\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0021","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={8, 9, 10}、E={{8, 9}}，邊無方向、沒有自環或重邊。候選端點集合 {8, 11} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{8, 11}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044021,"parameters":{"V":[8,9,10],"E":[[8,9]],"operation":"edge_membership","u":8,"v":11,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[8,9]],\"V\":[8,9,10],\"operation\":\"edge_membership\",\"u\":8,\"v\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0022","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={9, 10, 11, 12}、E={{9, 10}, {9, 11}, {10, 11}}，邊無方向、沒有自環或重邊。候選端點集合 {10, 9} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"是","explanation_zh":"{10, 9}∈E=是。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044022,"parameters":{"V":[9,10,11,12],"E":[[9,10],[9,11],[10,11]],"operation":"edge_membership","u":10,"v":9,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[9,10],[9,11],[10,11]],\"V\":[9,10,11,12],\"operation\":\"edge_membership\",\"u\":10,\"v\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0023","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={10, 11, 12, 13, 14}、E={{10, 11}, {12, 13}}，邊無方向、沒有自環或重邊。候選端點集合 {10, 14} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{10, 14}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044023,"parameters":{"V":[10,11,12,13,14],"E":[[10,11],[12,13]],"operation":"edge_membership","u":10,"v":14,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[10,11],[12,13]],\"V\":[10,11,12,13,14],\"operation\":\"edge_membership\",\"u\":10,\"v\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0024","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={11, 12, 13, 14, 15, 16}、E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}，邊無方向、沒有自環或重邊。候選端點集合 {11, 17} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{11, 17}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044024,"parameters":{"V":[11,12,13,14,15,16],"E":[[11,12],[11,13],[11,14],[11,15],[11,16],[12,13],[12,14],[12,15],[12,16],[13,14],[13,15],[13,16],[14,15],[14,16],[15,16]],"operation":"edge_membership","u":11,"v":17,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[11,12],[11,13],[11,14],[11,15],[11,16],[12,13],[12,14],[12,15],[12,16],[13,14],[13,15],[13,16],[14,15],[14,16],[15,16]],\"V\":[11,12,13,14,15,16],\"operation\":\"edge_membership\",\"u\":11,\"v\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0025","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的邊成員","stem_zh":"有限簡單無向圖 V={12, 13, 14}、E=∅，邊無方向、沒有自環或重邊。候選端點集合 {14, 12} 是否是一條 E 中的邊？","answer":{"kind":"short_text","value":"否","explanation_zh":"{14, 12}∈E=否。端點排序不改變無向邊；相同端點或域外端點都不能成為這個簡單圖的邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的邊成員"]},"variant":{"family_id":"qf-simple-graph-edge-membership","seed":202610044025,"parameters":{"V":[12,13,14],"E":[],"operation":"edge_membership","u":14,"v":12,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[],\"V\":[12,13,14],\"operation\":\"edge_membership\",\"u\":14,\"v\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0026","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={0}、E=∅，邊無方向、沒有自環或重邊。頂點 v=0 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"∅","explanation_zh":"N(0)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044026,"parameters":{"V":[0],"E":[],"operation":"neighbors","v":0,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[],\"V\":[0],\"operation\":\"neighbors\",\"v\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0027","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={-11, -10, -9, -8}、E={{-11, -10}, {-10, -9}, {-9, -8}}，邊無方向、沒有自環或重邊。頂點 v=-9 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{-10, -8}","explanation_zh":"N(-9)={-10, -8}；deg=2。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044027,"parameters":{"V":[-11,-10,-9,-8],"E":[[-11,-10],[-10,-9],[-9,-8]],"operation":"neighbors","v":-9,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-11,-10],[-10,-9],[-9,-8]],\"V\":[-11,-10,-9,-8],\"operation\":\"neighbors\",\"v\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0028","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={-10, -9, -8, -7, -6}、E={{-10, -9}, {-10, -8}, {-10, -7}}，邊無方向、沒有自環或重邊。頂點 v=-7 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{-10}","explanation_zh":"N(-7)={-10}；deg=1。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044028,"parameters":{"V":[-10,-9,-8,-7,-6],"E":[[-10,-9],[-10,-8],[-10,-7]],"operation":"neighbors","v":-7,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-10,-9],[-10,-8],[-10,-7]],\"V\":[-10,-9,-8,-7,-6],\"operation\":\"neighbors\",\"v\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0029","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={-9, -8, -7, -6, -5, -4}、E={{-9, -8}, {-9, -7}, {-8, -7}}，邊無方向、沒有自環或重邊。頂點 v=-5 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"∅","explanation_zh":"N(-5)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044029,"parameters":{"V":[-9,-8,-7,-6,-5,-4],"E":[[-9,-8],[-9,-7],[-8,-7]],"operation":"neighbors","v":-5,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-9,-8],[-9,-7],[-8,-7]],\"V\":[-9,-8,-7,-6,-5,-4],\"operation\":\"neighbors\",\"v\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0030","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={-8, -7, -6}、E={{-8, -7}}，邊無方向、沒有自環或重邊。頂點 v=-6 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"∅","explanation_zh":"N(-6)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044030,"parameters":{"V":[-8,-7,-6],"E":[[-8,-7]],"operation":"neighbors","v":-6,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-8,-7]],\"V\":[-8,-7,-6],\"operation\":\"neighbors\",\"v\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0031","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={-7, -6, -5, -4}、E={{-7, -6}, {-7, -5}, {-7, -4}, {-6, -5}, {-6, -4}, {-5, -4}}，邊無方向、沒有自環或重邊。頂點 v=-5 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{-7, -6, -4}","explanation_zh":"N(-5)={-7, -6, -4}；deg=3。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044031,"parameters":{"V":[-7,-6,-5,-4],"E":[[-7,-6],[-7,-5],[-7,-4],[-6,-5],[-6,-4],[-5,-4]],"operation":"neighbors","v":-5,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-7,-6],[-7,-5],[-7,-4],[-6,-5],[-6,-4],[-5,-4]],\"V\":[-7,-6,-5,-4],\"operation\":\"neighbors\",\"v\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0032","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={-6, -5, -4, -3, -2}、E=∅，邊無方向、沒有自環或重邊。頂點 v=-4 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"∅","explanation_zh":"N(-4)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044032,"parameters":{"V":[-6,-5,-4,-3,-2],"E":[],"operation":"neighbors","v":-4,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[],\"V\":[-6,-5,-4,-3,-2],\"operation\":\"neighbors\",\"v\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0033","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={-5, -4, -3, -2, -1, 0}、E={{-5, -4}, {-4, -3}, {-3, -2}, {-2, -1}, {-1, 0}}，邊無方向、沒有自環或重邊。頂點 v=-3 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{-4, -2}","explanation_zh":"N(-3)={-4, -2}；deg=2。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044033,"parameters":{"V":[-5,-4,-3,-2,-1,0],"E":[[-5,-4],[-4,-3],[-3,-2],[-2,-1],[-1,0]],"operation":"neighbors","v":-3,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-5,-4],[-4,-3],[-3,-2],[-2,-1],[-1,0]],\"V\":[-5,-4,-3,-2,-1,0],\"operation\":\"neighbors\",\"v\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0034","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={-4, -3, -2}、E={{-4, -3}}，邊無方向、沒有自環或重邊。頂點 v=-4 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{-3}","explanation_zh":"N(-4)={-3}；deg=1。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044034,"parameters":{"V":[-4,-3,-2],"E":[[-4,-3]],"operation":"neighbors","v":-4,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-4,-3]],\"V\":[-4,-3,-2],\"operation\":\"neighbors\",\"v\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0035","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={-3, -2, -1, 0}、E={{-3, -2}, {-3, -1}, {-2, -1}}，邊無方向、沒有自環或重邊。頂點 v=-1 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{-3, -2}","explanation_zh":"N(-1)={-3, -2}；deg=2。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044035,"parameters":{"V":[-3,-2,-1,0],"E":[[-3,-2],[-3,-1],[-2,-1]],"operation":"neighbors","v":-1,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-3,-2],[-3,-1],[-2,-1]],\"V\":[-3,-2,-1,0],\"operation\":\"neighbors\",\"v\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0036","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={-2, -1, 0, 1, 2}、E={{-2, -1}, {0, 1}}，邊無方向、沒有自環或重邊。頂點 v=-1 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{-2}","explanation_zh":"N(-1)={-2}；deg=1。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044036,"parameters":{"V":[-2,-1,0,1,2],"E":[[-2,-1],[0,1]],"operation":"neighbors","v":-1,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-2,-1],[0,1]],\"V\":[-2,-1,0,1,2],\"operation\":\"neighbors\",\"v\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0037","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={-1, 0, 1, 2, 3, 4}、E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}，邊無方向、沒有自環或重邊。頂點 v=-1 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{0, 1, 2, 3, 4}","explanation_zh":"N(-1)={0, 1, 2, 3, 4}；deg=5。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044037,"parameters":{"V":[-1,0,1,2,3,4],"E":[[-1,0],[-1,1],[-1,2],[-1,3],[-1,4],[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],"operation":"neighbors","v":-1,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-1,0],[-1,1],[-1,2],[-1,3],[-1,4],[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],\"V\":[-1,0,1,2,3,4],\"operation\":\"neighbors\",\"v\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0038","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={0, 1, 2}、E=∅，邊無方向、沒有自環或重邊。頂點 v=1 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"∅","explanation_zh":"N(1)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044038,"parameters":{"V":[0,1,2],"E":[],"operation":"neighbors","v":1,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[],\"V\":[0,1,2],\"operation\":\"neighbors\",\"v\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0039","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={1, 2, 3, 4}、E={{1, 2}, {2, 3}, {3, 4}}，邊無方向、沒有自環或重邊。頂點 v=3 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{2, 4}","explanation_zh":"N(3)={2, 4}；deg=2。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044039,"parameters":{"V":[1,2,3,4],"E":[[1,2],[2,3],[3,4]],"operation":"neighbors","v":3,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[1,2],[2,3],[3,4]],\"V\":[1,2,3,4],\"operation\":\"neighbors\",\"v\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0040","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={2, 3, 4, 5, 6}、E={{2, 3}, {2, 4}, {2, 5}}，邊無方向、沒有自環或重邊。頂點 v=2 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{3, 4, 5}","explanation_zh":"N(2)={3, 4, 5}；deg=3。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044040,"parameters":{"V":[2,3,4,5,6],"E":[[2,3],[2,4],[2,5]],"operation":"neighbors","v":2,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[2,3],[2,4],[2,5]],\"V\":[2,3,4,5,6],\"operation\":\"neighbors\",\"v\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0041","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={3, 4, 5, 6, 7, 8}、E={{3, 4}, {3, 5}, {4, 5}}，邊無方向、沒有自環或重邊。頂點 v=7 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"∅","explanation_zh":"N(7)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044041,"parameters":{"V":[3,4,5,6,7,8],"E":[[3,4],[3,5],[4,5]],"operation":"neighbors","v":7,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[3,4],[3,5],[4,5]],\"V\":[3,4,5,6,7,8],\"operation\":\"neighbors\",\"v\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0042","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={4, 5, 6}、E={{4, 5}}，邊無方向、沒有自環或重邊。頂點 v=6 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"∅","explanation_zh":"N(6)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044042,"parameters":{"V":[4,5,6],"E":[[4,5]],"operation":"neighbors","v":6,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[4,5]],\"V\":[4,5,6],\"operation\":\"neighbors\",\"v\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0043","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={5, 6, 7, 8}、E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}，邊無方向、沒有自環或重邊。頂點 v=7 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{5, 6, 8}","explanation_zh":"N(7)={5, 6, 8}；deg=3。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044043,"parameters":{"V":[5,6,7,8],"E":[[5,6],[5,7],[5,8],[6,7],[6,8],[7,8]],"operation":"neighbors","v":7,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[5,6],[5,7],[5,8],[6,7],[6,8],[7,8]],\"V\":[5,6,7,8],\"operation\":\"neighbors\",\"v\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0044","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={6, 7, 8, 9, 10}、E=∅，邊無方向、沒有自環或重邊。頂點 v=10 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"∅","explanation_zh":"N(10)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044044,"parameters":{"V":[6,7,8,9,10],"E":[],"operation":"neighbors","v":10,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[],\"V\":[6,7,8,9,10],\"operation\":\"neighbors\",\"v\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0045","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={7, 8, 9, 10, 11, 12}、E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}，邊無方向、沒有自環或重邊。頂點 v=9 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{8, 10}","explanation_zh":"N(9)={8, 10}；deg=2。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044045,"parameters":{"V":[7,8,9,10,11,12],"E":[[7,8],[8,9],[9,10],[10,11],[11,12]],"operation":"neighbors","v":9,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[7,8],[8,9],[9,10],[10,11],[11,12]],\"V\":[7,8,9,10,11,12],\"operation\":\"neighbors\",\"v\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0046","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={8, 9, 10}、E={{8, 9}}，邊無方向、沒有自環或重邊。頂點 v=8 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{9}","explanation_zh":"N(8)={9}；deg=1。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044046,"parameters":{"V":[8,9,10],"E":[[8,9]],"operation":"neighbors","v":8,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[8,9]],\"V\":[8,9,10],\"operation\":\"neighbors\",\"v\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0047","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={9, 10, 11, 12}、E={{9, 10}, {9, 11}, {10, 11}}，邊無方向、沒有自環或重邊。頂點 v=11 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{9, 10}","explanation_zh":"N(11)={9, 10}；deg=2。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044047,"parameters":{"V":[9,10,11,12],"E":[[9,10],[9,11],[10,11]],"operation":"neighbors","v":11,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[9,10],[9,11],[10,11]],\"V\":[9,10,11,12],\"operation\":\"neighbors\",\"v\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0048","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={10, 11, 12, 13, 14}、E={{10, 11}, {12, 13}}，邊無方向、沒有自環或重邊。頂點 v=13 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{12}","explanation_zh":"N(13)={12}；deg=1。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044048,"parameters":{"V":[10,11,12,13,14],"E":[[10,11],[12,13]],"operation":"neighbors","v":13,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[10,11],[12,13]],\"V\":[10,11,12,13,14],\"operation\":\"neighbors\",\"v\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0049","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={11, 12, 13, 14, 15, 16}、E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}，邊無方向、沒有自環或重邊。頂點 v=11 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"{12, 13, 14, 15, 16}","explanation_zh":"N(11)={12, 13, 14, 15, 16}；deg=5。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044049,"parameters":{"V":[11,12,13,14,15,16],"E":[[11,12],[11,13],[11,14],[11,15],[11,16],[12,13],[12,14],[12,15],[12,16],[13,14],[13,15],[13,16],[14,15],[14,16],[15,16]],"operation":"neighbors","v":11,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[11,12],[11,13],[11,14],[11,15],[11,16],[12,13],[12,14],[12,15],[12,16],[13,14],[13,15],[13,16],[14,15],[14,16],[15,16]],\"V\":[11,12,13,14,15,16],\"operation\":\"neighbors\",\"v\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0050","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的鄰居集合","stem_zh":"有限簡單無向圖 V={12, 13, 14}、E=∅，邊無方向、沒有自環或重邊。頂點 v=13 已屬於 V，求遞增且不重複的鄰居集合 N(v)。","answer":{"kind":"expression","value":"∅","explanation_zh":"N(13)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的鄰居集合"]},"variant":{"family_id":"qf-simple-graph-neighbors","seed":202610044050,"parameters":{"V":[12,13,14],"E":[],"operation":"neighbors","v":13,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[],\"V\":[12,13,14],\"operation\":\"neighbors\",\"v\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0051","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={0}、E=∅，邊無方向、沒有自環或重邊。頂點 v=0 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"0","explanation_zh":"N(0)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044051,"parameters":{"V":[0],"E":[],"operation":"degree","v":0,"semantic_template":"qf-simple-graph-degree/{\"E\":[],\"V\":[0],\"operation\":\"degree\",\"v\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0052","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={-11, -10, -9, -8}、E={{-11, -10}, {-10, -9}, {-9, -8}}，邊無方向、沒有自環或重邊。頂點 v=-9 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"2","explanation_zh":"N(-9)={-10, -8}；deg=2。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044052,"parameters":{"V":[-11,-10,-9,-8],"E":[[-11,-10],[-10,-9],[-9,-8]],"operation":"degree","v":-9,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-11,-10],[-10,-9],[-9,-8]],\"V\":[-11,-10,-9,-8],\"operation\":\"degree\",\"v\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0053","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={-10, -9, -8, -7, -6}、E={{-10, -9}, {-10, -8}, {-10, -7}}，邊無方向、沒有自環或重邊。頂點 v=-7 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"1","explanation_zh":"N(-7)={-10}；deg=1。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044053,"parameters":{"V":[-10,-9,-8,-7,-6],"E":[[-10,-9],[-10,-8],[-10,-7]],"operation":"degree","v":-7,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-10,-9],[-10,-8],[-10,-7]],\"V\":[-10,-9,-8,-7,-6],\"operation\":\"degree\",\"v\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0054","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={-9, -8, -7, -6, -5, -4}、E={{-9, -8}, {-9, -7}, {-8, -7}}，邊無方向、沒有自環或重邊。頂點 v=-5 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"0","explanation_zh":"N(-5)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044054,"parameters":{"V":[-9,-8,-7,-6,-5,-4],"E":[[-9,-8],[-9,-7],[-8,-7]],"operation":"degree","v":-5,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-9,-8],[-9,-7],[-8,-7]],\"V\":[-9,-8,-7,-6,-5,-4],\"operation\":\"degree\",\"v\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0055","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={-8, -7, -6}、E={{-8, -7}}，邊無方向、沒有自環或重邊。頂點 v=-6 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"0","explanation_zh":"N(-6)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044055,"parameters":{"V":[-8,-7,-6],"E":[[-8,-7]],"operation":"degree","v":-6,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-8,-7]],\"V\":[-8,-7,-6],\"operation\":\"degree\",\"v\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0056","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={-7, -6, -5, -4}、E={{-7, -6}, {-7, -5}, {-7, -4}, {-6, -5}, {-6, -4}, {-5, -4}}，邊無方向、沒有自環或重邊。頂點 v=-5 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"3","explanation_zh":"N(-5)={-7, -6, -4}；deg=3。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044056,"parameters":{"V":[-7,-6,-5,-4],"E":[[-7,-6],[-7,-5],[-7,-4],[-6,-5],[-6,-4],[-5,-4]],"operation":"degree","v":-5,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-7,-6],[-7,-5],[-7,-4],[-6,-5],[-6,-4],[-5,-4]],\"V\":[-7,-6,-5,-4],\"operation\":\"degree\",\"v\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0057","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={-6, -5, -4, -3, -2}、E=∅，邊無方向、沒有自環或重邊。頂點 v=-4 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"0","explanation_zh":"N(-4)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044057,"parameters":{"V":[-6,-5,-4,-3,-2],"E":[],"operation":"degree","v":-4,"semantic_template":"qf-simple-graph-degree/{\"E\":[],\"V\":[-6,-5,-4,-3,-2],\"operation\":\"degree\",\"v\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0058","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={-5, -4, -3, -2, -1, 0}、E={{-5, -4}, {-4, -3}, {-3, -2}, {-2, -1}, {-1, 0}}，邊無方向、沒有自環或重邊。頂點 v=-3 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"2","explanation_zh":"N(-3)={-4, -2}；deg=2。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044058,"parameters":{"V":[-5,-4,-3,-2,-1,0],"E":[[-5,-4],[-4,-3],[-3,-2],[-2,-1],[-1,0]],"operation":"degree","v":-3,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-5,-4],[-4,-3],[-3,-2],[-2,-1],[-1,0]],\"V\":[-5,-4,-3,-2,-1,0],\"operation\":\"degree\",\"v\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0059","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={-4, -3, -2}、E={{-4, -3}}，邊無方向、沒有自環或重邊。頂點 v=-4 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"1","explanation_zh":"N(-4)={-3}；deg=1。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044059,"parameters":{"V":[-4,-3,-2],"E":[[-4,-3]],"operation":"degree","v":-4,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-4,-3]],\"V\":[-4,-3,-2],\"operation\":\"degree\",\"v\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0060","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={-3, -2, -1, 0}、E={{-3, -2}, {-3, -1}, {-2, -1}}，邊無方向、沒有自環或重邊。頂點 v=-1 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"2","explanation_zh":"N(-1)={-3, -2}；deg=2。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044060,"parameters":{"V":[-3,-2,-1,0],"E":[[-3,-2],[-3,-1],[-2,-1]],"operation":"degree","v":-1,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-3,-2],[-3,-1],[-2,-1]],\"V\":[-3,-2,-1,0],\"operation\":\"degree\",\"v\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0061","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={-2, -1, 0, 1, 2}、E={{-2, -1}, {0, 1}}，邊無方向、沒有自環或重邊。頂點 v=-1 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"1","explanation_zh":"N(-1)={-2}；deg=1。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044061,"parameters":{"V":[-2,-1,0,1,2],"E":[[-2,-1],[0,1]],"operation":"degree","v":-1,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-2,-1],[0,1]],\"V\":[-2,-1,0,1,2],\"operation\":\"degree\",\"v\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0062","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={-1, 0, 1, 2, 3, 4}、E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}，邊無方向、沒有自環或重邊。頂點 v=-1 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"5","explanation_zh":"N(-1)={0, 1, 2, 3, 4}；deg=5。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044062,"parameters":{"V":[-1,0,1,2,3,4],"E":[[-1,0],[-1,1],[-1,2],[-1,3],[-1,4],[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],"operation":"degree","v":-1,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-1,0],[-1,1],[-1,2],[-1,3],[-1,4],[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],\"V\":[-1,0,1,2,3,4],\"operation\":\"degree\",\"v\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0063","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={0, 1, 2}、E=∅，邊無方向、沒有自環或重邊。頂點 v=1 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"0","explanation_zh":"N(1)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044063,"parameters":{"V":[0,1,2],"E":[],"operation":"degree","v":1,"semantic_template":"qf-simple-graph-degree/{\"E\":[],\"V\":[0,1,2],\"operation\":\"degree\",\"v\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0064","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={1, 2, 3, 4}、E={{1, 2}, {2, 3}, {3, 4}}，邊無方向、沒有自環或重邊。頂點 v=3 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"2","explanation_zh":"N(3)={2, 4}；deg=2。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044064,"parameters":{"V":[1,2,3,4],"E":[[1,2],[2,3],[3,4]],"operation":"degree","v":3,"semantic_template":"qf-simple-graph-degree/{\"E\":[[1,2],[2,3],[3,4]],\"V\":[1,2,3,4],\"operation\":\"degree\",\"v\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0065","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={2, 3, 4, 5, 6}、E={{2, 3}, {2, 4}, {2, 5}}，邊無方向、沒有自環或重邊。頂點 v=2 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"3","explanation_zh":"N(2)={3, 4, 5}；deg=3。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044065,"parameters":{"V":[2,3,4,5,6],"E":[[2,3],[2,4],[2,5]],"operation":"degree","v":2,"semantic_template":"qf-simple-graph-degree/{\"E\":[[2,3],[2,4],[2,5]],\"V\":[2,3,4,5,6],\"operation\":\"degree\",\"v\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0066","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={3, 4, 5, 6, 7, 8}、E={{3, 4}, {3, 5}, {4, 5}}，邊無方向、沒有自環或重邊。頂點 v=7 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"0","explanation_zh":"N(7)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044066,"parameters":{"V":[3,4,5,6,7,8],"E":[[3,4],[3,5],[4,5]],"operation":"degree","v":7,"semantic_template":"qf-simple-graph-degree/{\"E\":[[3,4],[3,5],[4,5]],\"V\":[3,4,5,6,7,8],\"operation\":\"degree\",\"v\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0067","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={4, 5, 6}、E={{4, 5}}，邊無方向、沒有自環或重邊。頂點 v=6 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"0","explanation_zh":"N(6)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044067,"parameters":{"V":[4,5,6],"E":[[4,5]],"operation":"degree","v":6,"semantic_template":"qf-simple-graph-degree/{\"E\":[[4,5]],\"V\":[4,5,6],\"operation\":\"degree\",\"v\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0068","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={5, 6, 7, 8}、E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}，邊無方向、沒有自環或重邊。頂點 v=7 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"3","explanation_zh":"N(7)={5, 6, 8}；deg=3。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044068,"parameters":{"V":[5,6,7,8],"E":[[5,6],[5,7],[5,8],[6,7],[6,8],[7,8]],"operation":"degree","v":7,"semantic_template":"qf-simple-graph-degree/{\"E\":[[5,6],[5,7],[5,8],[6,7],[6,8],[7,8]],\"V\":[5,6,7,8],\"operation\":\"degree\",\"v\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0069","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={6, 7, 8, 9, 10}、E=∅，邊無方向、沒有自環或重邊。頂點 v=10 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"0","explanation_zh":"N(10)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044069,"parameters":{"V":[6,7,8,9,10],"E":[],"operation":"degree","v":10,"semantic_template":"qf-simple-graph-degree/{\"E\":[],\"V\":[6,7,8,9,10],\"operation\":\"degree\",\"v\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0070","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={7, 8, 9, 10, 11, 12}、E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}，邊無方向、沒有自環或重邊。頂點 v=9 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"2","explanation_zh":"N(9)={8, 10}；deg=2。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044070,"parameters":{"V":[7,8,9,10,11,12],"E":[[7,8],[8,9],[9,10],[10,11],[11,12]],"operation":"degree","v":9,"semantic_template":"qf-simple-graph-degree/{\"E\":[[7,8],[8,9],[9,10],[10,11],[11,12]],\"V\":[7,8,9,10,11,12],\"operation\":\"degree\",\"v\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0071","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={8, 9, 10}、E={{8, 9}}，邊無方向、沒有自環或重邊。頂點 v=8 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"1","explanation_zh":"N(8)={9}；deg=1。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044071,"parameters":{"V":[8,9,10],"E":[[8,9]],"operation":"degree","v":8,"semantic_template":"qf-simple-graph-degree/{\"E\":[[8,9]],\"V\":[8,9,10],\"operation\":\"degree\",\"v\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0072","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={9, 10, 11, 12}、E={{9, 10}, {9, 11}, {10, 11}}，邊無方向、沒有自環或重邊。頂點 v=11 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"2","explanation_zh":"N(11)={9, 10}；deg=2。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044072,"parameters":{"V":[9,10,11,12],"E":[[9,10],[9,11],[10,11]],"operation":"degree","v":11,"semantic_template":"qf-simple-graph-degree/{\"E\":[[9,10],[9,11],[10,11]],\"V\":[9,10,11,12],\"operation\":\"degree\",\"v\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0073","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={10, 11, 12, 13, 14}、E={{10, 11}, {12, 13}}，邊無方向、沒有自環或重邊。頂點 v=13 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"1","explanation_zh":"N(13)={12}；deg=1。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044073,"parameters":{"V":[10,11,12,13,14],"E":[[10,11],[12,13]],"operation":"degree","v":13,"semantic_template":"qf-simple-graph-degree/{\"E\":[[10,11],[12,13]],\"V\":[10,11,12,13,14],\"operation\":\"degree\",\"v\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0074","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={11, 12, 13, 14, 15, 16}、E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}，邊無方向、沒有自環或重邊。頂點 v=11 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"5","explanation_zh":"N(11)={12, 13, 14, 15, 16}；deg=5。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044074,"parameters":{"V":[11,12,13,14,15,16],"E":[[11,12],[11,13],[11,14],[11,15],[11,16],[12,13],[12,14],[12,15],[12,16],[13,14],[13,15],[13,16],[14,15],[14,16],[15,16]],"operation":"degree","v":11,"semantic_template":"qf-simple-graph-degree/{\"E\":[[11,12],[11,13],[11,14],[11,15],[11,16],[12,13],[12,14],[12,15],[12,16],[13,14],[13,15],[13,16],[14,15],[14,16],[15,16]],\"V\":[11,12,13,14,15,16],\"operation\":\"degree\",\"v\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0075","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的頂點度數","stem_zh":"有限簡單無向圖 V={12, 13, 14}、E=∅，邊無方向、沒有自環或重邊。頂點 v=13 已屬於 V，求頂點度數 deg(v)。","answer":{"kind":"numeric","value":"0","explanation_zh":"N(13)=∅；deg=0。按明示邊集找不同鄰居，不由示意圖或線條交叉猜相鄰性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的頂點度數"]},"variant":{"family_id":"qf-simple-graph-degree","seed":202610044075,"parameters":{"V":[12,13,14],"E":[],"operation":"degree","v":13,"semantic_template":"qf-simple-graph-degree/{\"E\":[],\"V\":[12,13,14],\"operation\":\"degree\",\"v\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0076","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V=∅、E=∅，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"度數和=0；2|E|=0。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044076,"parameters":{"V":[],"E":[],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[],\"V\":[],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0077","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={-11, -10, -9, -8}、E={{-11, -10}, {-10, -9}, {-9, -8}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"6","explanation_zh":"度數和=6；2|E|=6。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044077,"parameters":{"V":[-11,-10,-9,-8],"E":[[-11,-10],[-10,-9],[-9,-8]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-11,-10],[-10,-9],[-9,-8]],\"V\":[-11,-10,-9,-8],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0078","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={-10, -9, -8, -7, -6}、E={{-10, -9}, {-10, -8}, {-10, -7}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"6","explanation_zh":"度數和=6；2|E|=6。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044078,"parameters":{"V":[-10,-9,-8,-7,-6],"E":[[-10,-9],[-10,-8],[-10,-7]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-10,-9],[-10,-8],[-10,-7]],\"V\":[-10,-9,-8,-7,-6],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0079","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={-9, -8, -7, -6, -5, -4}、E={{-9, -8}, {-9, -7}, {-8, -7}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"6","explanation_zh":"度數和=6；2|E|=6。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044079,"parameters":{"V":[-9,-8,-7,-6,-5,-4],"E":[[-9,-8],[-9,-7],[-8,-7]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-9,-8],[-9,-7],[-8,-7]],\"V\":[-9,-8,-7,-6,-5,-4],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0080","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={-8, -7, -6}、E={{-8, -7}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"2","explanation_zh":"度數和=2；2|E|=2。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044080,"parameters":{"V":[-8,-7,-6],"E":[[-8,-7]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-8,-7]],\"V\":[-8,-7,-6],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0081","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={-7, -6, -5, -4}、E={{-7, -6}, {-7, -5}, {-7, -4}, {-6, -5}, {-6, -4}, {-5, -4}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"12","explanation_zh":"度數和=12；2|E|=12。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044081,"parameters":{"V":[-7,-6,-5,-4],"E":[[-7,-6],[-7,-5],[-7,-4],[-6,-5],[-6,-4],[-5,-4]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-7,-6],[-7,-5],[-7,-4],[-6,-5],[-6,-4],[-5,-4]],\"V\":[-7,-6,-5,-4],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0082","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={-6, -5, -4, -3, -2}、E=∅，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"度數和=0；2|E|=0。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044082,"parameters":{"V":[-6,-5,-4,-3,-2],"E":[],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[],\"V\":[-6,-5,-4,-3,-2],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0083","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={-5, -4, -3, -2, -1, 0}、E={{-5, -4}, {-4, -3}, {-3, -2}, {-2, -1}, {-1, 0}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"10","explanation_zh":"度數和=10；2|E|=10。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044083,"parameters":{"V":[-5,-4,-3,-2,-1,0],"E":[[-5,-4],[-4,-3],[-3,-2],[-2,-1],[-1,0]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-5,-4],[-4,-3],[-3,-2],[-2,-1],[-1,0]],\"V\":[-5,-4,-3,-2,-1,0],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0084","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={-4, -3, -2}、E={{-4, -3}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"2","explanation_zh":"度數和=2；2|E|=2。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044084,"parameters":{"V":[-4,-3,-2],"E":[[-4,-3]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-4,-3]],\"V\":[-4,-3,-2],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0085","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={-3, -2, -1, 0}、E={{-3, -2}, {-3, -1}, {-2, -1}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"6","explanation_zh":"度數和=6；2|E|=6。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044085,"parameters":{"V":[-3,-2,-1,0],"E":[[-3,-2],[-3,-1],[-2,-1]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-3,-2],[-3,-1],[-2,-1]],\"V\":[-3,-2,-1,0],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0086","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={-2, -1, 0, 1, 2}、E={{-2, -1}, {0, 1}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"4","explanation_zh":"度數和=4；2|E|=4。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044086,"parameters":{"V":[-2,-1,0,1,2],"E":[[-2,-1],[0,1]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-2,-1],[0,1]],\"V\":[-2,-1,0,1,2],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0087","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={-1, 0, 1, 2, 3, 4}、E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"30","explanation_zh":"度數和=30；2|E|=30。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044087,"parameters":{"V":[-1,0,1,2,3,4],"E":[[-1,0],[-1,1],[-1,2],[-1,3],[-1,4],[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-1,0],[-1,1],[-1,2],[-1,3],[-1,4],[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],\"V\":[-1,0,1,2,3,4],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0088","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={0, 1, 2}、E=∅，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"度數和=0；2|E|=0。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044088,"parameters":{"V":[0,1,2],"E":[],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[],\"V\":[0,1,2],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0089","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={1, 2, 3, 4}、E={{1, 2}, {2, 3}, {3, 4}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"6","explanation_zh":"度數和=6；2|E|=6。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044089,"parameters":{"V":[1,2,3,4],"E":[[1,2],[2,3],[3,4]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[1,2],[2,3],[3,4]],\"V\":[1,2,3,4],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0090","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={2, 3, 4, 5, 6}、E={{2, 3}, {2, 4}, {2, 5}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"6","explanation_zh":"度數和=6；2|E|=6。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044090,"parameters":{"V":[2,3,4,5,6],"E":[[2,3],[2,4],[2,5]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[2,3],[2,4],[2,5]],\"V\":[2,3,4,5,6],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0091","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={3, 4, 5, 6, 7, 8}、E={{3, 4}, {3, 5}, {4, 5}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"6","explanation_zh":"度數和=6；2|E|=6。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044091,"parameters":{"V":[3,4,5,6,7,8],"E":[[3,4],[3,5],[4,5]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[3,4],[3,5],[4,5]],\"V\":[3,4,5,6,7,8],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0092","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={4, 5, 6}、E={{4, 5}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"2","explanation_zh":"度數和=2；2|E|=2。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044092,"parameters":{"V":[4,5,6],"E":[[4,5]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[4,5]],\"V\":[4,5,6],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0093","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={5, 6, 7, 8}、E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"12","explanation_zh":"度數和=12；2|E|=12。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044093,"parameters":{"V":[5,6,7,8],"E":[[5,6],[5,7],[5,8],[6,7],[6,8],[7,8]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[5,6],[5,7],[5,8],[6,7],[6,8],[7,8]],\"V\":[5,6,7,8],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0094","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={6, 7, 8, 9, 10}、E=∅，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"度數和=0；2|E|=0。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044094,"parameters":{"V":[6,7,8,9,10],"E":[],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[],\"V\":[6,7,8,9,10],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0095","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={7, 8, 9, 10, 11, 12}、E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"10","explanation_zh":"度數和=10；2|E|=10。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044095,"parameters":{"V":[7,8,9,10,11,12],"E":[[7,8],[8,9],[9,10],[10,11],[11,12]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[7,8],[8,9],[9,10],[10,11],[11,12]],\"V\":[7,8,9,10,11,12],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0096","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={8, 9, 10}、E={{8, 9}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"2","explanation_zh":"度數和=2；2|E|=2。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044096,"parameters":{"V":[8,9,10],"E":[[8,9]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[8,9]],\"V\":[8,9,10],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0097","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={9, 10, 11, 12}、E={{9, 10}, {9, 11}, {10, 11}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"6","explanation_zh":"度數和=6；2|E|=6。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044097,"parameters":{"V":[9,10,11,12],"E":[[9,10],[9,11],[10,11]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[9,10],[9,11],[10,11]],\"V\":[9,10,11,12],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0098","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={10, 11, 12, 13, 14}、E={{10, 11}, {12, 13}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"4","explanation_zh":"度數和=4；2|E|=4。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044098,"parameters":{"V":[10,11,12,13,14],"E":[[10,11],[12,13]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[10,11],[12,13]],\"V\":[10,11,12,13,14],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0099","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={11, 12, 13, 14, 15, 16}、E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"30","explanation_zh":"度數和=30；2|E|=30。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044099,"parameters":{"V":[11,12,13,14,15,16],"E":[[11,12],[11,13],[11,14],[11,15],[11,16],[12,13],[12,14],[12,15],[12,16],[13,14],[13,15],[13,16],[14,15],[14,16],[15,16]],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[[11,12],[11,13],[11,14],[11,15],[11,16],[12,13],[12,14],[12,15],[12,16],[13,14],[13,15],[13,16],[14,15],[14,16],[15,16]],\"V\":[11,12,13,14,15,16],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r04-graph-0100","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"簡單無向圖的全頂點度數和","stem_zh":"有限簡單無向圖 V={12, 13, 14}、E=∅，邊無方向、沒有自環或重邊。全部 V 中頂點的度數和是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"度數和=0；2|E|=0。每條邊在兩個端點各貢獻一次，孤立頂點貢獻零；空圖總和也是零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["簡單無向圖的全頂點度數和"]},"variant":{"family_id":"qf-simple-graph-handshaking","seed":202610044100,"parameters":{"V":[12,13,14],"E":[],"operation":"handshaking","semantic_template":"qf-simple-graph-handshaking/{\"E\":[],\"V\":[12,13,14],\"operation\":\"handshaking\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}