{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0001","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(0)=∅，無向邊列項=[]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(0, [], [], [], 0, 0)","explanation_zh":"離散結構核對=(0, [], [], [], 0, 0)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175001,"parameters":{"n":0,"edges":[],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0002","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(1)=[0]，無向邊列項=[]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(1, [], [[否]], [0], 0, 0)","explanation_zh":"離散結構核對=(1, [], [[否]], [0], 0, 0)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175002,"parameters":{"n":1,"edges":[],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0003","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[1, 2], [1, 3], [2, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(4, [[1, 2], [1, 3], [2, 3]], [[否, 否, 否, 否], [否, 否, 是, 是], [否, 是, 否, 是], [否, 是, 是, 否]], [0, 2, 2, 2], 6, 6)","explanation_zh":"離散結構核對=(4, [[1, 2], [1, 3], [2, 3]], [[否, 否, 否, 否], [否, 否, 是, 是], [否, 是, 否, 是], [否, 是, 是, 否]], [0, 2, 2, 2], 6, 6)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175003,"parameters":{"n":4,"edges":[[1,2],[1,3],[2,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[1,2],[1,3],[2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0004","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[0, 1], [2, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(4, [[0, 1], [2, 3]], [[否, 是, 否, 否], [是, 否, 否, 否], [否, 否, 否, 是], [否, 否, 是, 否]], [1, 1, 1, 1], 4, 4)","explanation_zh":"離散結構核對=(4, [[0, 1], [2, 3]], [[否, 是, 否, 否], [是, 否, 否, 否], [否, 否, 否, 是], [否, 否, 是, 否]], [1, 1, 1, 1], 4, 4)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175004,"parameters":{"n":4,"edges":[[0,1],[2,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0005","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[0, 1], [1, 2], [2, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(4, [[0, 1], [1, 2], [2, 3]], [[否, 是, 否, 否], [是, 否, 是, 否], [否, 是, 否, 是], [否, 否, 是, 否]], [1, 2, 2, 1], 6, 6)","explanation_zh":"離散結構核對=(4, [[0, 1], [1, 2], [2, 3]], [[否, 是, 否, 否], [是, 否, 是, 否], [否, 是, 否, 是], [否, 否, 是, 否]], [1, 2, 2, 1], 6, 6)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175005,"parameters":{"n":4,"edges":[[0,1],[1,2],[2,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[1,2],[2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0006","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[0, 1], [0, 2], [0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(4, [[0, 1], [0, 2], [0, 3]], [[否, 是, 是, 是], [是, 否, 否, 否], [是, 否, 否, 否], [是, 否, 否, 否]], [3, 1, 1, 1], 6, 6)","explanation_zh":"離散結構核對=(4, [[0, 1], [0, 2], [0, 3]], [[否, 是, 是, 是], [是, 否, 否, 否], [是, 否, 否, 否], [是, 否, 否, 否]], [3, 1, 1, 1], 6, 6)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175006,"parameters":{"n":4,"edges":[[0,1],[0,2],[0,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0007","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 1]], [[否, 是, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [1, 1, 0, 0, 0], 2, 2)","explanation_zh":"離散結構核對=(5, [[0, 1]], [[否, 是, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [1, 1, 0, 0, 0], 2, 2)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175007,"parameters":{"n":5,"edges":[[0,1]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0008","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 2]], [[否, 否, 是, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [1, 0, 1, 0, 0], 2, 2)","explanation_zh":"離散結構核對=(5, [[0, 2]], [[否, 否, 是, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [1, 0, 1, 0, 0], 2, 2)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175008,"parameters":{"n":5,"edges":[[0,2]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0009","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2]], [[否, 是, 是, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [2, 1, 1, 0, 0], 4, 4)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2]], [[否, 是, 是, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [2, 1, 1, 0, 0], 4, 4)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175009,"parameters":{"n":5,"edges":[[0,1],[0,2]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0010","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 3]], [[否, 否, 否, 是, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [1, 0, 0, 1, 0], 2, 2)","explanation_zh":"離散結構核對=(5, [[0, 3]], [[否, 否, 否, 是, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [1, 0, 0, 1, 0], 2, 2)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175010,"parameters":{"n":5,"edges":[[0,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0011","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 3]], [[否, 是, 否, 是, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [2, 1, 0, 1, 0], 4, 4)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 3]], [[否, 是, 否, 是, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [2, 1, 0, 1, 0], 4, 4)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175011,"parameters":{"n":5,"edges":[[0,1],[0,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0012","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 3]], [[否, 否, 是, 是, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [2, 0, 1, 1, 0], 4, 4)","explanation_zh":"離散結構核對=(5, [[0, 2], [0, 3]], [[否, 否, 是, 是, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [2, 0, 1, 1, 0], 4, 4)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175012,"parameters":{"n":5,"edges":[[0,2],[0,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,2],[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0013","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 3]], [[否, 是, 是, 是, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [3, 1, 1, 1, 0], 6, 6)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2], [0, 3]], [[否, 是, 是, 是, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [3, 1, 1, 1, 0], 6, 6)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175013,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0014","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 4]], [[否, 否, 否, 否, 是], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [1, 0, 0, 0, 1], 2, 2)","explanation_zh":"離散結構核對=(5, [[0, 4]], [[否, 否, 否, 否, 是], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [1, 0, 0, 0, 1], 2, 2)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175014,"parameters":{"n":5,"edges":[[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0015","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 4]], [[否, 是, 否, 否, 是], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [2, 1, 0, 0, 1], 4, 4)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 4]], [[否, 是, 否, 否, 是], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [2, 1, 0, 0, 1], 4, 4)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175015,"parameters":{"n":5,"edges":[[0,1],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0016","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 4]], [[否, 否, 是, 否, 是], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [2, 0, 1, 0, 1], 4, 4)","explanation_zh":"離散結構核對=(5, [[0, 2], [0, 4]], [[否, 否, 是, 否, 是], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [2, 0, 1, 0, 1], 4, 4)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175016,"parameters":{"n":5,"edges":[[0,2],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,2],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0017","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 4]], [[否, 是, 是, 否, 是], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [3, 1, 1, 0, 1], 6, 6)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2], [0, 4]], [[否, 是, 是, 否, 是], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [3, 1, 1, 0, 1], 6, 6)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175017,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,2],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0018","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 3], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 3], [0, 4]], [[否, 否, 否, 是, 是], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [2, 0, 0, 1, 1], 4, 4)","explanation_zh":"離散結構核對=(5, [[0, 3], [0, 4]], [[否, 否, 否, 是, 是], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [2, 0, 0, 1, 1], 4, 4)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175018,"parameters":{"n":5,"edges":[[0,3],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0019","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 3], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 3], [0, 4]], [[否, 是, 否, 是, 是], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [3, 1, 0, 1, 1], 6, 6)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 3], [0, 4]], [[否, 是, 否, 是, 是], [是, 否, 否, 否, 否], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [3, 1, 0, 1, 1], 6, 6)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175019,"parameters":{"n":5,"edges":[[0,1],[0,3],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0020","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 3], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 3], [0, 4]], [[否, 否, 是, 是, 是], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [3, 0, 1, 1, 1], 6, 6)","explanation_zh":"離散結構核對=(5, [[0, 2], [0, 3], [0, 4]], [[否, 否, 是, 是, 是], [否, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [3, 0, 1, 1, 1], 6, 6)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175020,"parameters":{"n":5,"edges":[[0,2],[0,3],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,2],[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0021","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 3], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[否, 是, 是, 是, 是], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [4, 1, 1, 1, 1], 8, 8)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[否, 是, 是, 是, 是], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否], [是, 否, 否, 否, 否]], [4, 1, 1, 1, 1], 8, 8)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175021,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,3],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,2],[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0022","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[1, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[1, 2]], [[否, 否, 否, 否, 否], [否, 否, 是, 否, 否], [否, 是, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [0, 1, 1, 0, 0], 2, 2)","explanation_zh":"離散結構核對=(5, [[1, 2]], [[否, 否, 否, 否, 否], [否, 否, 是, 否, 否], [否, 是, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [0, 1, 1, 0, 0], 2, 2)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175022,"parameters":{"n":5,"edges":[[1,2]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0023","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2]], [[否, 是, 否, 否, 否], [是, 否, 是, 否, 否], [否, 是, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [1, 2, 1, 0, 0], 4, 4)","explanation_zh":"離散結構核對=(5, [[0, 1], [1, 2]], [[否, 是, 否, 否, 否], [是, 否, 是, 否, 否], [否, 是, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [1, 2, 1, 0, 0], 4, 4)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175023,"parameters":{"n":5,"edges":[[0,1],[1,2]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0024","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [1, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 2], [1, 2]], [[否, 否, 是, 否, 否], [否, 否, 是, 否, 否], [是, 是, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [1, 1, 2, 0, 0], 4, 4)","explanation_zh":"離散結構核對=(5, [[0, 2], [1, 2]], [[否, 否, 是, 否, 否], [否, 否, 是, 否, 否], [是, 是, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [1, 1, 2, 0, 0], 4, 4)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175024,"parameters":{"n":5,"edges":[[0,2],[1,2]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,2],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0025","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"完整鄰接矩陣與握手數","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [1, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 按全部頂點索引的布林鄰接矩陣, 度數向量, 度數和, 兩倍邊數)。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2]], [[否, 是, 是, 否, 否], [是, 否, 是, 否, 否], [是, 是, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [2, 2, 2, 0, 0], 6, 6)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2], [1, 2]], [[否, 是, 是, 否, 否], [是, 否, 是, 否, 否], [是, 是, 否, 否, 否], [否, 否, 否, 否, 否], [否, 否, 否, 否, 否]], [2, 2, 2, 0, 0], 6, 6)；每條無向邊加入兩個鄰居位置，再逐頂點計不同鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完整鄰接矩陣與握手數"]},"variant":{"family_id":"qf-density17-graph-adjacency-degrees","seed":202610175025,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2]],"domain":"Fin_simple_undirected_graph","operation":"graph-adjacency-degrees","semantic_template":"qf-density17-graph-adjacency-degrees/{\"edges\":[[0,1],[0,2],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0026","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(0)=∅，無向邊列項=[]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(0, [], [], [], [])","explanation_zh":"離散結構核對=(0, [], [], [], [])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175026,"parameters":{"n":0,"edges":[],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0027","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(1)=[0]，無向邊列項=[]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(1, [], [[]], [[]], [0])","explanation_zh":"離散結構核對=(1, [], [[]], [[]], [0])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175027,"parameters":{"n":1,"edges":[],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0028","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[1, 2], [1, 3], [2, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(4, [[1, 2], [1, 3], [2, 3]], [[], [2, 3], [1, 3], [1, 2]], [[1, 2, 3], [0], [0], [0]], [0])","explanation_zh":"離散結構核對=(4, [[1, 2], [1, 3], [2, 3]], [[], [2, 3], [1, 3], [1, 2]], [[1, 2, 3], [0], [0], [0]], [0])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175028,"parameters":{"n":4,"edges":[[1,2],[1,3],[2,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[1,2],[1,3],[2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0029","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[0, 1], [2, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(4, [[0, 1], [2, 3]], [[1], [0], [3], [2]], [[2, 3], [2, 3], [0, 1], [0, 1]], [])","explanation_zh":"離散結構核對=(4, [[0, 1], [2, 3]], [[1], [0], [3], [2]], [[2, 3], [2, 3], [0, 1], [0, 1]], [])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175029,"parameters":{"n":4,"edges":[[0,1],[2,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0030","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[0, 1], [1, 2], [2, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(4, [[0, 1], [1, 2], [2, 3]], [[1], [0, 2], [1, 3], [2]], [[2, 3], [3], [0], [0, 1]], [])","explanation_zh":"離散結構核對=(4, [[0, 1], [1, 2], [2, 3]], [[1], [0, 2], [1, 3], [2]], [[2, 3], [3], [0], [0, 1]], [])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175030,"parameters":{"n":4,"edges":[[0,1],[1,2],[2,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[1,2],[2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0031","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[0, 1], [0, 2], [0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(4, [[0, 1], [0, 2], [0, 3]], [[1, 2, 3], [0], [0], [0]], [[], [2, 3], [1, 3], [1, 2]], [])","explanation_zh":"離散結構核對=(4, [[0, 1], [0, 2], [0, 3]], [[1, 2, 3], [0], [0], [0]], [[], [2, 3], [1, 3], [1, 2]], [])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175031,"parameters":{"n":4,"edges":[[0,1],[0,2],[0,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0032","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 1]], [[1], [0], [], [], []], [[2, 3, 4], [2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [2, 3, 4])","explanation_zh":"離散結構核對=(5, [[0, 1]], [[1], [0], [], [], []], [[2, 3, 4], [2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [2, 3, 4])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175032,"parameters":{"n":5,"edges":[[0,1]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0033","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 2]], [[2], [], [0], [], []], [[1, 3, 4], [0, 2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [1, 3, 4])","explanation_zh":"離散結構核對=(5, [[0, 2]], [[2], [], [0], [], []], [[1, 3, 4], [0, 2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [1, 3, 4])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175033,"parameters":{"n":5,"edges":[[0,2]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0034","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2]], [[1, 2], [0], [0], [], []], [[3, 4], [2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2]], [[1, 2], [0], [0], [], []], [[3, 4], [2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175034,"parameters":{"n":5,"edges":[[0,1],[0,2]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0035","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 3]], [[3], [], [], [0], []], [[1, 2, 4], [0, 2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [1, 2, 4])","explanation_zh":"離散結構核對=(5, [[0, 3]], [[3], [], [], [0], []], [[1, 2, 4], [0, 2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [1, 2, 4])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175035,"parameters":{"n":5,"edges":[[0,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0036","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 3]], [[1, 3], [0], [], [0], []], [[2, 4], [2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [2, 4])","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 3]], [[1, 3], [0], [], [0], []], [[2, 4], [2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [2, 4])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175036,"parameters":{"n":5,"edges":[[0,1],[0,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0037","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 3]], [[2, 3], [], [0], [0], []], [[1, 4], [0, 2, 3, 4], [1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [1, 4])","explanation_zh":"離散結構核對=(5, [[0, 2], [0, 3]], [[2, 3], [], [0], [0], []], [[1, 4], [0, 2, 3, 4], [1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [1, 4])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175037,"parameters":{"n":5,"edges":[[0,2],[0,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,2],[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0038","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 3]], [[1, 2, 3], [0], [0], [0], []], [[4], [2, 3, 4], [1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [4])","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2], [0, 3]], [[1, 2, 3], [0], [0], [0], []], [[4], [2, 3, 4], [1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [4])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175038,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,3]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0039","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 4]], [[4], [], [], [], [0]], [[1, 2, 3], [0, 2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [1, 2, 3])","explanation_zh":"離散結構核對=(5, [[0, 4]], [[4], [], [], [], [0]], [[1, 2, 3], [0, 2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [1, 2, 3])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175039,"parameters":{"n":5,"edges":[[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0040","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 4]], [[1, 4], [0], [], [], [0]], [[2, 3], [2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [2, 3])","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 4]], [[1, 4], [0], [], [], [0]], [[2, 3], [2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [2, 3])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175040,"parameters":{"n":5,"edges":[[0,1],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0041","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 4]], [[2, 4], [], [0], [], [0]], [[1, 3], [0, 2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [1, 3])","explanation_zh":"離散結構核對=(5, [[0, 2], [0, 4]], [[2, 4], [], [0], [], [0]], [[1, 3], [0, 2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [1, 3])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175041,"parameters":{"n":5,"edges":[[0,2],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,2],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0042","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 4]], [[1, 2, 4], [0], [0], [], [0]], [[3], [2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [3])","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2], [0, 4]], [[1, 2, 4], [0], [0], [], [0]], [[3], [2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [3])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175042,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,2],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0043","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 3], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 3], [0, 4]], [[3, 4], [], [], [0], [0]], [[1, 2], [0, 2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [1, 2, 3]], [1, 2])","explanation_zh":"離散結構核對=(5, [[0, 3], [0, 4]], [[3, 4], [], [], [0], [0]], [[1, 2], [0, 2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [1, 2, 3]], [1, 2])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175043,"parameters":{"n":5,"edges":[[0,3],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0044","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 3], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 3], [0, 4]], [[1, 3, 4], [0], [], [0], [0]], [[2], [2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [1, 2, 3]], [2])","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 3], [0, 4]], [[1, 3, 4], [0], [], [0], [0]], [[2], [2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [1, 2, 3]], [2])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175044,"parameters":{"n":5,"edges":[[0,1],[0,3],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0045","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 3], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 3], [0, 4]], [[2, 3, 4], [], [0], [0], [0]], [[1], [0, 2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]], [1])","explanation_zh":"離散結構核對=(5, [[0, 2], [0, 3], [0, 4]], [[2, 3, 4], [], [0], [0], [0]], [[1], [0, 2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]], [1])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175045,"parameters":{"n":5,"edges":[[0,2],[0,3],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,2],[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0046","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 3], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[1, 2, 3, 4], [0], [0], [0], [0]], [[], [2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]], [])","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[1, 2, 3, 4], [0], [0], [0], [0]], [[], [2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]], [])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175046,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,3],[0,4]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,2],[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0047","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[1, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[1, 2]], [[], [2], [1], [], []], [[1, 2, 3, 4], [0, 3, 4], [0, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [0, 3, 4])","explanation_zh":"離散結構核對=(5, [[1, 2]], [[], [2], [1], [], []], [[1, 2, 3, 4], [0, 3, 4], [0, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [0, 3, 4])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175047,"parameters":{"n":5,"edges":[[1,2]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0048","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2]], [[1], [0, 2], [1], [], []], [[2, 3, 4], [3, 4], [0, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])","explanation_zh":"離散結構核對=(5, [[0, 1], [1, 2]], [[1], [0, 2], [1], [], []], [[2, 3, 4], [3, 4], [0, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175048,"parameters":{"n":5,"edges":[[0,1],[1,2]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0049","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [1, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 2], [1, 2]], [[2], [2], [0, 1], [], []], [[1, 3, 4], [0, 3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])","explanation_zh":"離散結構核對=(5, [[0, 2], [1, 2]], [[2], [2], [0, 1], [], []], [[1, 3, 4], [0, 3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175049,"parameters":{"n":5,"edges":[[0,2],[1,2]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,2],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0050","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"每頂點鄰居與非鄰居","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [1, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2]], [[1, 2], [0, 2], [0, 1], [], []], [[3, 4], [3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2], [1, 2]], [[1, 2], [0, 2], [0, 1], [], []], [[3, 4], [3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["每頂點鄰居與非鄰居"]},"variant":{"family_id":"qf-density17-graph-complete-neighbor-partitions","seed":202610175050,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2]],"domain":"Fin_simple_undirected_graph","operation":"graph-complete-neighbor-partitions","semantic_template":"qf-density17-graph-complete-neighbor-partitions/{\"edges\":[[0,1],[0,2],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0051","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(2)=[0, 1]，無向邊列項=[[0, 1]]；各邊兩端皆合法且相異。候選邊=[1, 0]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1]], [0, 1], 移除, [], [1, 1], [0, 0], [-1, -1], -2)","explanation_zh":"離散結構核對=([[0, 1]], [0, 1], 移除, [], [1, 1], [0, 0], [-1, -1], -2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175051,"parameters":{"n":2,"edges":[[0,1]],"edge":[1,0],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,1],\"edges\":[[0,1]],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0052","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(2)=[0, 1]，無向邊列項=[]；各邊兩端皆合法且相異。候選邊=[0, 1]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([], [0, 1], 新增, [[0, 1]], [0, 0], [1, 1], [1, 1], 2)","explanation_zh":"離散結構核對=([], [0, 1], 新增, [[0, 1]], [0, 0], [1, 1], [1, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175052,"parameters":{"n":2,"edges":[],"edge":[0,1],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,1],\"edges\":[],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0053","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(3)=[0, 1, 2]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。候選邊=[2, 0]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1], [1, 2]], [0, 2], 新增, [[0, 1], [0, 2], [1, 2]], [1, 2, 1], [2, 2, 2], [1, 0, 1], 2)","explanation_zh":"離散結構核對=([[0, 1], [1, 2]], [0, 2], 新增, [[0, 1], [0, 2], [1, 2]], [1, 2, 1], [2, 2, 2], [1, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175053,"parameters":{"n":3,"edges":[[0,1],[1,2]],"edge":[2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,2],\"edges\":[[0,1],[1,2]],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0054","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(3)=[0, 1, 2]，無向邊列項=[[0, 1], [0, 2]]；各邊兩端皆合法且相異。候選邊=[0, 1]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1], [0, 2]], [0, 1], 移除, [[0, 2]], [2, 1, 1], [1, 0, 1], [-1, -1, 0], -2)","explanation_zh":"離散結構核對=([[0, 1], [0, 2]], [0, 1], 移除, [[0, 2]], [2, 1, 1], [1, 0, 1], [-1, -1, 0], -2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175054,"parameters":{"n":3,"edges":[[0,1],[0,2]],"edge":[0,1],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,1],\"edges\":[[0,1],[0,2]],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0055","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1]], [0, 4], 新增, [[0, 1], [0, 4]], [1, 1, 0, 0, 0], [2, 1, 0, 0, 1], [1, 0, 0, 0, 1], 2)","explanation_zh":"離散結構核對=([[0, 1]], [0, 4], 新增, [[0, 1], [0, 4]], [1, 1, 0, 0, 0], [2, 1, 0, 0, 1], [1, 0, 0, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175055,"parameters":{"n":5,"edges":[[0,1]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0056","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 2]], [0, 4], 新增, [[0, 2], [0, 4]], [1, 0, 1, 0, 0], [2, 0, 1, 0, 1], [1, 0, 0, 0, 1], 2)","explanation_zh":"離散結構核對=([[0, 2]], [0, 4], 新增, [[0, 2], [0, 4]], [1, 0, 1, 0, 0], [2, 0, 1, 0, 1], [1, 0, 0, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175056,"parameters":{"n":5,"edges":[[0,2]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0057","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1], [0, 2]], [0, 4], 新增, [[0, 1], [0, 2], [0, 4]], [2, 1, 1, 0, 0], [3, 1, 1, 0, 1], [1, 0, 0, 0, 1], 2)","explanation_zh":"離散結構核對=([[0, 1], [0, 2]], [0, 4], 新增, [[0, 1], [0, 2], [0, 4]], [2, 1, 1, 0, 0], [3, 1, 1, 0, 1], [1, 0, 0, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175057,"parameters":{"n":5,"edges":[[0,1],[0,2]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0058","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 3]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 3]], [0, 4], 新增, [[0, 3], [0, 4]], [1, 0, 0, 1, 0], [2, 0, 0, 1, 1], [1, 0, 0, 0, 1], 2)","explanation_zh":"離散結構核對=([[0, 3]], [0, 4], 新增, [[0, 3], [0, 4]], [1, 0, 0, 1, 0], [2, 0, 0, 1, 1], [1, 0, 0, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175058,"parameters":{"n":5,"edges":[[0,3]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0059","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 3]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1], [0, 3]], [0, 4], 新增, [[0, 1], [0, 3], [0, 4]], [2, 1, 0, 1, 0], [3, 1, 0, 1, 1], [1, 0, 0, 0, 1], 2)","explanation_zh":"離散結構核對=([[0, 1], [0, 3]], [0, 4], 新增, [[0, 1], [0, 3], [0, 4]], [2, 1, 0, 1, 0], [3, 1, 0, 1, 1], [1, 0, 0, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175059,"parameters":{"n":5,"edges":[[0,1],[0,3]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0060","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 3]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 2], [0, 3]], [0, 4], 新增, [[0, 2], [0, 3], [0, 4]], [2, 0, 1, 1, 0], [3, 0, 1, 1, 1], [1, 0, 0, 0, 1], 2)","explanation_zh":"離散結構核對=([[0, 2], [0, 3]], [0, 4], 新增, [[0, 2], [0, 3], [0, 4]], [2, 0, 1, 1, 0], [3, 0, 1, 1, 1], [1, 0, 0, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175060,"parameters":{"n":5,"edges":[[0,2],[0,3]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,2],[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0061","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 3]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 3]], [0, 4], 新增, [[0, 1], [0, 2], [0, 3], [0, 4]], [3, 1, 1, 1, 0], [4, 1, 1, 1, 1], [1, 0, 0, 0, 1], 2)","explanation_zh":"離散結構核對=([[0, 1], [0, 2], [0, 3]], [0, 4], 新增, [[0, 1], [0, 2], [0, 3], [0, 4]], [3, 1, 1, 1, 0], [4, 1, 1, 1, 1], [1, 0, 0, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175061,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,3]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,2],[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0062","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 4]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 4]], [0, 4], 移除, [], [1, 0, 0, 0, 1], [0, 0, 0, 0, 0], [-1, 0, 0, 0, -1], -2)","explanation_zh":"離散結構核對=([[0, 4]], [0, 4], 移除, [], [1, 0, 0, 0, 1], [0, 0, 0, 0, 0], [-1, 0, 0, 0, -1], -2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175062,"parameters":{"n":5,"edges":[[0,4]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0063","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 4]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1], [0, 4]], [0, 4], 移除, [[0, 1]], [2, 1, 0, 0, 1], [1, 1, 0, 0, 0], [-1, 0, 0, 0, -1], -2)","explanation_zh":"離散結構核對=([[0, 1], [0, 4]], [0, 4], 移除, [[0, 1]], [2, 1, 0, 0, 1], [1, 1, 0, 0, 0], [-1, 0, 0, 0, -1], -2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175063,"parameters":{"n":5,"edges":[[0,1],[0,4]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0064","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 4]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 2], [0, 4]], [0, 4], 移除, [[0, 2]], [2, 0, 1, 0, 1], [1, 0, 1, 0, 0], [-1, 0, 0, 0, -1], -2)","explanation_zh":"離散結構核對=([[0, 2], [0, 4]], [0, 4], 移除, [[0, 2]], [2, 0, 1, 0, 1], [1, 0, 1, 0, 0], [-1, 0, 0, 0, -1], -2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175064,"parameters":{"n":5,"edges":[[0,2],[0,4]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,2],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0065","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 4]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 4]], [0, 4], 移除, [[0, 1], [0, 2]], [3, 1, 1, 0, 1], [2, 1, 1, 0, 0], [-1, 0, 0, 0, -1], -2)","explanation_zh":"離散結構核對=([[0, 1], [0, 2], [0, 4]], [0, 4], 移除, [[0, 1], [0, 2]], [3, 1, 1, 0, 1], [2, 1, 1, 0, 0], [-1, 0, 0, 0, -1], -2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175065,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,4]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,2],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0066","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 3], [0, 4]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 3], [0, 4]], [0, 4], 移除, [[0, 3]], [2, 0, 0, 1, 1], [1, 0, 0, 1, 0], [-1, 0, 0, 0, -1], -2)","explanation_zh":"離散結構核對=([[0, 3], [0, 4]], [0, 4], 移除, [[0, 3]], [2, 0, 0, 1, 1], [1, 0, 0, 1, 0], [-1, 0, 0, 0, -1], -2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175066,"parameters":{"n":5,"edges":[[0,3],[0,4]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0067","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 3], [0, 4]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1], [0, 3], [0, 4]], [0, 4], 移除, [[0, 1], [0, 3]], [3, 1, 0, 1, 1], [2, 1, 0, 1, 0], [-1, 0, 0, 0, -1], -2)","explanation_zh":"離散結構核對=([[0, 1], [0, 3], [0, 4]], [0, 4], 移除, [[0, 1], [0, 3]], [3, 1, 0, 1, 1], [2, 1, 0, 1, 0], [-1, 0, 0, 0, -1], -2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175067,"parameters":{"n":5,"edges":[[0,1],[0,3],[0,4]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0068","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 3], [0, 4]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 2], [0, 3], [0, 4]], [0, 4], 移除, [[0, 2], [0, 3]], [3, 0, 1, 1, 1], [2, 0, 1, 1, 0], [-1, 0, 0, 0, -1], -2)","explanation_zh":"離散結構核對=([[0, 2], [0, 3], [0, 4]], [0, 4], 移除, [[0, 2], [0, 3]], [3, 0, 1, 1, 1], [2, 0, 1, 1, 0], [-1, 0, 0, 0, -1], -2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175068,"parameters":{"n":5,"edges":[[0,2],[0,3],[0,4]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,2],[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0069","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 3], [0, 4]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 3], [0, 4]], [0, 4], 移除, [[0, 1], [0, 2], [0, 3]], [4, 1, 1, 1, 1], [3, 1, 1, 1, 0], [-1, 0, 0, 0, -1], -2)","explanation_zh":"離散結構核對=([[0, 1], [0, 2], [0, 3], [0, 4]], [0, 4], 移除, [[0, 1], [0, 2], [0, 3]], [4, 1, 1, 1, 1], [3, 1, 1, 1, 0], [-1, 0, 0, 0, -1], -2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175069,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,3],[0,4]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,2],[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0070","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[1, 2]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[1, 2]], [0, 4], 新增, [[0, 4], [1, 2]], [0, 1, 1, 0, 0], [1, 1, 1, 0, 1], [1, 0, 0, 0, 1], 2)","explanation_zh":"離散結構核對=([[1, 2]], [0, 4], 新增, [[0, 4], [1, 2]], [0, 1, 1, 0, 0], [1, 1, 1, 0, 1], [1, 0, 0, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175070,"parameters":{"n":5,"edges":[[1,2]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0071","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1], [1, 2]], [0, 4], 新增, [[0, 1], [0, 4], [1, 2]], [1, 2, 1, 0, 0], [2, 2, 1, 0, 1], [1, 0, 0, 0, 1], 2)","explanation_zh":"離散結構核對=([[0, 1], [1, 2]], [0, 4], 新增, [[0, 1], [0, 4], [1, 2]], [1, 2, 1, 0, 0], [2, 2, 1, 0, 1], [1, 0, 0, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175071,"parameters":{"n":5,"edges":[[0,1],[1,2]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0072","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [1, 2]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 2], [1, 2]], [0, 4], 新增, [[0, 2], [0, 4], [1, 2]], [1, 1, 2, 0, 0], [2, 1, 2, 0, 1], [1, 0, 0, 0, 1], 2)","explanation_zh":"離散結構核對=([[0, 2], [1, 2]], [0, 4], 新增, [[0, 2], [0, 4], [1, 2]], [1, 1, 2, 0, 0], [2, 1, 2, 0, 1], [1, 0, 0, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175072,"parameters":{"n":5,"edges":[[0,2],[1,2]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,2],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0073","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [1, 2]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [1, 2]], [0, 4], 新增, [[0, 1], [0, 2], [0, 4], [1, 2]], [2, 2, 2, 0, 0], [3, 2, 2, 0, 1], [1, 0, 0, 0, 1], 2)","explanation_zh":"離散結構核對=([[0, 1], [0, 2], [1, 2]], [0, 4], 新增, [[0, 1], [0, 2], [0, 4], [1, 2]], [2, 2, 2, 0, 0], [3, 2, 2, 0, 1], [1, 0, 0, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175073,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,2],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0074","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 3], [1, 2]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 3], [1, 2]], [0, 4], 新增, [[0, 3], [0, 4], [1, 2]], [1, 1, 1, 1, 0], [2, 1, 1, 1, 1], [1, 0, 0, 0, 1], 2)","explanation_zh":"離散結構核對=([[0, 3], [1, 2]], [0, 4], 新增, [[0, 3], [0, 4], [1, 2]], [1, 1, 1, 1, 0], [2, 1, 1, 1, 1], [1, 0, 0, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175074,"parameters":{"n":5,"edges":[[0,3],[1,2]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,3],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0075","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"切換一條邊的有號度差","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 3], [1, 2]]；各邊兩端皆合法且相異。候選邊=[0, 4]。已存在則移除，未存在則新增。答 (原邊集, 排序候選邊, 新增或移除, 新邊集, 原度數, 新度數, 有號度數差, 度差總和)。","answer":{"kind":"expression","value":"([[0, 1], [0, 3], [1, 2]], [0, 4], 新增, [[0, 1], [0, 3], [0, 4], [1, 2]], [2, 2, 1, 1, 0], [3, 2, 1, 1, 1], [1, 0, 0, 0, 1], 2)","explanation_zh":"離散結構核對=([[0, 1], [0, 3], [1, 2]], [0, 4], 新增, [[0, 1], [0, 3], [0, 4], [1, 2]], [2, 2, 1, 1, 0], [3, 2, 1, 1, 1], [1, 0, 0, 0, 1], 2)；先判是否原有此無向邊，再改一條邊並逐頂點計新度數。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["切換一條邊的有號度差"]},"variant":{"family_id":"qf-density17-graph-edge-toggle-delta","seed":202610175075,"parameters":{"n":5,"edges":[[0,1],[0,3],[1,2]],"edge":[0,4],"domain":"Fin_simple_undirected_graph","operation":"graph-edge-toggle-delta","semantic_template":"qf-density17-graph-edge-toggle-delta/{\"edge\":[0,4],\"edges\":[[0,1],[0,3],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0076","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(0)=∅，無向邊列項=[]；各邊兩端皆合法且相異。頂點子集列項=[]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([], [], [], [], [])","explanation_zh":"離散結構核對=([], [], [], [], [])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175076,"parameters":{"n":0,"edges":[],"subset":[],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[],\"n\":0,\"subset\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0077","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(3)=[0, 1, 2]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。頂點子集列項=[]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1], [1, 2]], [], [], [], [])","explanation_zh":"離散結構核對=([[0, 1], [1, 2]], [], [], [], [])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175077,"parameters":{"n":3,"edges":[[0,1],[1,2]],"subset":[],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[1,2]],\"n\":3,\"subset\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0078","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(3)=[0, 1, 2]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。頂點子集列項=[0, 2]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1], [1, 2]], [0, 2], [], [0, 0], [[], []])","explanation_zh":"離散結構核對=([[0, 1], [1, 2]], [0, 2], [], [0, 0], [[], []])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175078,"parameters":{"n":3,"edges":[[0,1],[1,2]],"subset":[0,2],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[1,2]],\"n\":3,\"subset\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0079","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(3)=[0, 1, 2]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。頂點子集列項=[2, 1]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1], [1, 2]], [1, 2], [[1, 2]], [1, 1], [[2], [1]])","explanation_zh":"離散結構核對=([[0, 1], [1, 2]], [1, 2], [[1, 2]], [1, 1], [[2], [1]])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175079,"parameters":{"n":3,"edges":[[0,1],[1,2]],"subset":[2,1],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[1,2]],\"n\":3,\"subset\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0080","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[0, 1], [2, 3]]；各邊兩端皆合法且相異。頂點子集列項=[3, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [0, 2, 3], [[2, 3]], [0, 1, 1], [[], [3], [2]])","explanation_zh":"離散結構核對=([[0, 1], [2, 3]], [0, 2, 3], [[2, 3]], [0, 1, 1], [[], [3], [2]])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175080,"parameters":{"n":4,"edges":[[0,1],[2,3]],"subset":[3,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[2,3]],\"n\":4,\"subset\":[0,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0081","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1]], [0, 2, 4], [], [0, 0, 0], [[], [], []])","explanation_zh":"離散結構核對=([[0, 1]], [0, 2, 4], [], [0, 0, 0], [[], [], []])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175081,"parameters":{"n":5,"edges":[[0,1]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0082","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])","explanation_zh":"離散結構核對=([[0, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175082,"parameters":{"n":5,"edges":[[0,2]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,2]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0083","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1], [0, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])","explanation_zh":"離散結構核對=([[0, 1], [0, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175083,"parameters":{"n":5,"edges":[[0,1],[0,2]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,2]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0084","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 3]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 3]], [0, 2, 4], [], [0, 0, 0], [[], [], []])","explanation_zh":"離散結構核對=([[0, 3]], [0, 2, 4], [], [0, 0, 0], [[], [], []])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175084,"parameters":{"n":5,"edges":[[0,3]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,3]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0085","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 3]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1], [0, 3]], [0, 2, 4], [], [0, 0, 0], [[], [], []])","explanation_zh":"離散結構核對=([[0, 1], [0, 3]], [0, 2, 4], [], [0, 0, 0], [[], [], []])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175085,"parameters":{"n":5,"edges":[[0,1],[0,3]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,3]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0086","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 3]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 2], [0, 3]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])","explanation_zh":"離散結構核對=([[0, 2], [0, 3]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175086,"parameters":{"n":5,"edges":[[0,2],[0,3]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,2],[0,3]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0087","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 3]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 3]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])","explanation_zh":"離散結構核對=([[0, 1], [0, 2], [0, 3]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175087,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,3]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0088","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 4]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])","explanation_zh":"離散結構核對=([[0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175088,"parameters":{"n":5,"edges":[[0,4]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,4]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0089","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 4]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])","explanation_zh":"離散結構核對=([[0, 1], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175089,"parameters":{"n":5,"edges":[[0,1],[0,4]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,4]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0090","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 4]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 2], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])","explanation_zh":"離散結構核對=([[0, 2], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175090,"parameters":{"n":5,"edges":[[0,2],[0,4]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,2],[0,4]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0091","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 4]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])","explanation_zh":"離散結構核對=([[0, 1], [0, 2], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175091,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,4]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,2],[0,4]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0092","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 3], [0, 4]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 3], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])","explanation_zh":"離散結構核對=([[0, 3], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175092,"parameters":{"n":5,"edges":[[0,3],[0,4]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,3],[0,4]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0093","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 3], [0, 4]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1], [0, 3], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])","explanation_zh":"離散結構核對=([[0, 1], [0, 3], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175093,"parameters":{"n":5,"edges":[[0,1],[0,3],[0,4]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,3],[0,4]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0094","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 3], [0, 4]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 2], [0, 3], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])","explanation_zh":"離散結構核對=([[0, 2], [0, 3], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175094,"parameters":{"n":5,"edges":[[0,2],[0,3],[0,4]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,2],[0,3],[0,4]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0095","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 3], [0, 4]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [0, 3], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])","explanation_zh":"離散結構核對=([[0, 1], [0, 2], [0, 3], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175095,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,3],[0,4]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,2],[0,3],[0,4]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0096","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[1, 2]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])","explanation_zh":"離散結構核對=([[1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175096,"parameters":{"n":5,"edges":[[1,2]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[1,2]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0097","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1], [1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])","explanation_zh":"離散結構核對=([[0, 1], [1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175097,"parameters":{"n":5,"edges":[[0,1],[1,2]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[1,2]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0098","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [1, 2]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 2], [1, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])","explanation_zh":"離散結構核對=([[0, 2], [1, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175098,"parameters":{"n":5,"edges":[[0,2],[1,2]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,2],[1,2]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0099","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [1, 2]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 1], [0, 2], [1, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])","explanation_zh":"離散結構核對=([[0, 1], [0, 2], [1, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175099,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,1],[0,2],[1,2]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r05-graph-0100","batch_id":"batch-20261012-advance-20261017-r05-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-graph"],"title_zh":"保留原標籤的誘導子圖","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 3], [1, 2]]；各邊兩端皆合法且相異。頂點子集列項=[4, 2, 0]。答 (原正規邊集, 正規子集, 誘導邊集, 子集頂點依原標籤遞增的度向量, 同序鄰居列表)。不重新編號。","answer":{"kind":"expression","value":"([[0, 3], [1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])","explanation_zh":"離散結構核對=([[0, 3], [1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])；只保留兩端都在子集的邊，再以保留標籤計鄰居。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["保留原標籤的誘導子圖"]},"variant":{"family_id":"qf-density17-graph-induced-subset","seed":202610175100,"parameters":{"n":5,"edges":[[0,3],[1,2]],"subset":[4,2,0],"domain":"Fin_simple_undirected_graph","operation":"graph-induced-subset","semantic_template":"qf-density17-graph-induced-subset/{\"edges\":[[0,3],[1,2]],\"n\":5,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}