{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0001","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3], [[v0, v1], [v1, v2], [v2, v3]], [[v0, v2], [v1, v2], [v0, v1, v2], [v1, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3]], 2, [[v0, v2], [v1, v2], [v1, v3]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3], [[v0, v1], [v1, v2], [v2, v3]], [[v0, v2], [v1, v2], [v0, v1, v2], [v1, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3]], 2, [[v0, v2], [v1, v2], [v1, v3]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156001,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0002","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], [[v0, v1], [v1, v2], [v2, v3], [v3, v4]], [[v1, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3], [v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v1, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], 2, [[v1, v3]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4], [[v0, v1], [v1, v2], [v2, v3], [v3, v4]], [[v1, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3], [v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v1, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], 2, [[v1, v3]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156002,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0003","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [[v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v1, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4], [v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5]], 3, [[v0, v2, v4], [v1, v2, v4], [v1, v3, v4], [v1, v3, v5]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [[v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v1, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4], [v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5]], 3, [[v0, v2, v4], [v1, v2, v4], [v1, v3, v4], [v1, v3, v5]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156003,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0004","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [[v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6]], 3, [[v1, v3, v5]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [[v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6]], 3, [[v1, v3, v5]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156004,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0005","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [[v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 4, [[v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6], [v1, v3, v5, v7]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [[v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 4, [[v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6], [v1, v3, v5, v7]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156005,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0006","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [[v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 4, [[v1, v3, v5, v7]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [[v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 4, [[v1, v3, v5, v7]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156006,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0007","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8], [v1, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v1, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v1, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v1, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v1, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v1, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v1, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v1, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v1, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v1, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v1, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v1, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v1, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v1, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v1, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v1, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v1, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v1, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v1, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v1, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v1, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v1, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v1, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v1, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v1, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v1, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v1, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v1, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v1, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v1, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v1, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v1, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v1, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]], 5, [[v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8], [v1, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8], [v1, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v1, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v1, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v1, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v1, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v1, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v1, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v1, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v1, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v1, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v1, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v1, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v1, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v1, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v1, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v1, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v1, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v1, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v1, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v1, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v1, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v1, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v1, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v1, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v1, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v1, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v1, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v1, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v1, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v1, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v1, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v1, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v1, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]], 5, [[v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8], [v1, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156007,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0008","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v1, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v0, v1, v10, v3, v5, v6, v8], [v0, v10, v2, v3, v5, v6, v8], [v1, v10, v2, v3, v5, v6, v8], [v0, v1, v10, v2, v3, v5, v6, v8], [v0, v10, v2, v4, v5, v6, v8], [v1, v10, v2, v4, v5, v6, v8], [v0, v1, v10, v2, v4, v5, v6, v8], [v1, v10, v3, v4, v5, v6, v8], [v0, v1, v10, v3, v4, v5, v6, v8], [v0, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v10, v3, v5, v7, v8], [v0, v10, v2, v3, v5, v7, v8], [v1, v10, v2, v3, v5, v7, v8], [v0, v1, v10, v2, v3, v5, v7, v8], [v0, v10, v2, v4, v5, v7, v8], [v1, v10, v2, v4, v5, v7, v8], [v0, v1, v10, v2, v4, v5, v7, v8], [v1, v10, v3, v4, v5, v7, v8], [v0, v1, v10, v3, v4, v5, v7, v8], [v0, v10, v2, v3, v4, v5, v7, v8], [v1, v10, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v7, v8], [v0, v10, v2, v4, v6, v7, v8], [v1, v10, v2, v4, v6, v7, v8], [v0, v1, v10, v2, v4, v6, v7, v8], [v1, v10, v3, v4, v6, v7, v8], [v0, v1, v10, v3, v4, v6, v7, v8], [v0, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v3, v5, v6, v7, v8], [v0, v1, v10, v3, v5, v6, v7, v8], [v0, v10, v2, v3, v5, v6, v7, v8], [v1, v10, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v5, v6, v7, v8], [v0, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v4, v5, v6, v7, v8], [v0, v10, v2, v3, v4, v5, v6, v7, v8], [v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8], [v1, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v1, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v10, v2, v3, v5, v7, v9], [v1, v10, v2, v3, v5, v7, v9], [v0, v1, v10, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v1, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v0, v10, v2, v4, v5, v7, v9], [v1, v10, v2, v4, v5, v7, v9], [v0, v1, v10, v2, v4, v5, v7, v9], [v1, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v1, v10, v3, v4, v5, v7, v9], [v0, v1, v10, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v1, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v10, v2, v3, v4, v5, v7, v9], [v1, v10, v2, v3, v4, v5, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v1, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v0, v10, v2, v4, v6, v7, v9], [v1, v10, v2, v4, v6, v7, v9], [v0, v1, v10, v2, v4, v6, v7, v9], [v1, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v1, v10, v3, v4, v6, v7, v9], [v0, v1, v10, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v1, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v0, v10, v2, v3, v4, v6, v7, v9], [v1, v10, v2, v3, v4, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v9], [v1, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v1, v10, v3, v5, v6, v7, v9], [v0, v1, v10, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v1, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v10, v2, v3, v5, v6, v7, v9], [v1, v10, v2, v3, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v1, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v0, v10, v2, v4, v5, v6, v7, v9], [v1, v10, v2, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v9], [v1, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v1, v10, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v1, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v1, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v0, v10, v2, v4, v6, v8, v9], [v1, v10, v2, v4, v6, v8, v9], [v0, v1, v10, v2, v4, v6, v8, v9], [v1, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v1, v10, v3, v4, v6, v8, v9], [v0, v1, v10, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v1, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v0, v10, v2, v3, v4, v6, v8, v9], [v1, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v8, v9], [v1, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v1, v10, v3, v5, v6, v8, v9], [v0, v1, v10, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v1, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v10, v2, v3, v5, v6, v8, v9], [v1, v10, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v1, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v0, v10, v2, v4, v5, v6, v8, v9], [v1, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v8, v9], [v1, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v1, v10, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v1, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [v1, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v1, v10, v3, v5, v7, v8, v9], [v0, v1, v10, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v1, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v10, v2, v3, v5, v7, v8, v9], [v1, v10, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v1, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v0, v10, v2, v4, v5, v7, v8, v9], [v1, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v7, v8, v9], [v1, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v1, v10, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v1, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v1, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v0, v10, v2, v4, v6, v7, v8, v9], [v1, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v6, v7, v8, v9], [v1, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v1, v10, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v1, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v1, v10, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v1, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v1, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]], 5, [[v1, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v1, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v0, v1, v10, v3, v5, v6, v8], [v0, v10, v2, v3, v5, v6, v8], [v1, v10, v2, v3, v5, v6, v8], [v0, v1, v10, v2, v3, v5, v6, v8], [v0, v10, v2, v4, v5, v6, v8], [v1, v10, v2, v4, v5, v6, v8], [v0, v1, v10, v2, v4, v5, v6, v8], [v1, v10, v3, v4, v5, v6, v8], [v0, v1, v10, v3, v4, v5, v6, v8], [v0, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v10, v3, v5, v7, v8], [v0, v10, v2, v3, v5, v7, v8], [v1, v10, v2, v3, v5, v7, v8], [v0, v1, v10, v2, v3, v5, v7, v8], [v0, v10, v2, v4, v5, v7, v8], [v1, v10, v2, v4, v5, v7, v8], [v0, v1, v10, v2, v4, v5, v7, v8], [v1, v10, v3, v4, v5, v7, v8], [v0, v1, v10, v3, v4, v5, v7, v8], [v0, v10, v2, v3, v4, v5, v7, v8], [v1, v10, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v7, v8], [v0, v10, v2, v4, v6, v7, v8], [v1, v10, v2, v4, v6, v7, v8], [v0, v1, v10, v2, v4, v6, v7, v8], [v1, v10, v3, v4, v6, v7, v8], [v0, v1, v10, v3, v4, v6, v7, v8], [v0, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v3, v5, v6, v7, v8], [v0, v1, v10, v3, v5, v6, v7, v8], [v0, v10, v2, v3, v5, v6, v7, v8], [v1, v10, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v5, v6, v7, v8], [v0, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v4, v5, v6, v7, v8], [v0, v10, v2, v3, v4, v5, v6, v7, v8], [v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8], [v1, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v1, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v10, v2, v3, v5, v7, v9], [v1, v10, v2, v3, v5, v7, v9], [v0, v1, v10, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v1, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v0, v10, v2, v4, v5, v7, v9], [v1, v10, v2, v4, v5, v7, v9], [v0, v1, v10, v2, v4, v5, v7, v9], [v1, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v1, v10, v3, v4, v5, v7, v9], [v0, v1, v10, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v1, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v10, v2, v3, v4, v5, v7, v9], [v1, v10, v2, v3, v4, v5, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v1, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v0, v10, v2, v4, v6, v7, v9], [v1, v10, v2, v4, v6, v7, v9], [v0, v1, v10, v2, v4, v6, v7, v9], [v1, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v1, v10, v3, v4, v6, v7, v9], [v0, v1, v10, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v1, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v0, v10, v2, v3, v4, v6, v7, v9], [v1, v10, v2, v3, v4, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v9], [v1, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v1, v10, v3, v5, v6, v7, v9], [v0, v1, v10, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v1, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v10, v2, v3, v5, v6, v7, v9], [v1, v10, v2, v3, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v1, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v0, v10, v2, v4, v5, v6, v7, v9], [v1, v10, v2, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v9], [v1, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v1, v10, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v1, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v1, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v0, v10, v2, v4, v6, v8, v9], [v1, v10, v2, v4, v6, v8, v9], [v0, v1, v10, v2, v4, v6, v8, v9], [v1, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v1, v10, v3, v4, v6, v8, v9], [v0, v1, v10, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v1, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v0, v10, v2, v3, v4, v6, v8, v9], [v1, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v8, v9], [v1, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v1, v10, v3, v5, v6, v8, v9], [v0, v1, v10, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v1, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v10, v2, v3, v5, v6, v8, v9], [v1, v10, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v1, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v0, v10, v2, v4, v5, v6, v8, v9], [v1, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v8, v9], [v1, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v1, v10, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v1, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [v1, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v1, v10, v3, v5, v7, v8, v9], [v0, v1, v10, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v1, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v10, v2, v3, v5, v7, v8, v9], [v1, v10, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v1, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v0, v10, v2, v4, v5, v7, v8, v9], [v1, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v7, v8, v9], [v1, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v1, v10, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v1, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v1, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v0, v10, v2, v4, v6, v7, v8, v9], [v1, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v6, v7, v8, v9], [v1, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v1, v10, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v1, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v1, v10, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v1, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v1, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]], 5, [[v1, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156008,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0009","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v11\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v0, 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v11, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v1, v10, v3, v5, v7, v9], [v1, v11, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v0, v10, v11, v2, v4, v6, v8], [v1, v10, v11, v2, v4, v6, v8], [v0, v1, v10, v11, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v1, v10, v11, v3, v4, v6, v8], [v0, v1, v10, v11, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v1, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v0, v10, v11, v2, v3, v4, v6, v8], [v1, v10, v11, 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v6, v7, v8, v9], [v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v1, v10, v3, v5, v7, v9], [v1, v11, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156009,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v11"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0010","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3], [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], [[v0, v2], [v0, v1, v2], [v1, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3]], 2, [[v0, v2], [v1, v3]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3], [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], [[v0, v2], [v0, v1, v2], [v1, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3]], 2, [[v0, v2], [v1, v3]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156010,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0011","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], [[v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v1, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], 3, [[v0, v1, v3], [v0, v2, v3], [v0, v2, v4], [v1, v2, v4], [v1, v3, v4]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4], [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], [[v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v1, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], 3, [[v0, v1, v3], [v0, v2, v3], [v0, v2, v4], [v1, v2, v4], [v1, v3, v4]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156011,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0012","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [[v0, v2, v4], [v0, v1, v2, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5]], 3, [[v0, v2, v4], [v1, v3, v5]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [[v0, v2, v4], [v0, v1, v2, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5]], 3, [[v0, v2, v4], [v1, v3, v5]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156012,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0013","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6]], 4, [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v2, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6]], 4, [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v2, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156013,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v6\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0014","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [[v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 4, [[v0, v2, v4, v6], [v1, v3, v5, v7]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [[v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 4, [[v0, v2, v4, v6], [v1, v3, v5, v7]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156014,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v7\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0015","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156015,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v8\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0016","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8], [v1, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v1, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v1, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v1, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v1, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v1, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v1, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v1, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v1, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v1, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v1, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v1, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v1, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v1, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v1, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v1, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v1, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v1, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v1, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v1, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v1, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v1, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v1, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v1, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v1, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v1, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v1, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v1, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v1, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v1, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v1, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v1, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v1, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]], 5, [[v0, v2, v4, v6, v8], [v1, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8], [v1, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v1, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v1, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v1, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v1, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v1, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v1, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v1, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v1, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v1, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v1, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v1, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v1, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v1, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v1, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v1, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v1, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v1, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v1, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v1, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v1, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v1, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v1, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v1, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v1, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v1, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v1, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v1, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v1, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v1, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v1, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v1, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v1, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]], 5, [[v0, v2, v4, v6, v8], [v1, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156016,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v9\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0017","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v1, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v0, v1, v10, v3, v5, v6, v8], [v0, v10, v2, v3, v5, v6, v8], [v1, v10, v2, v3, v5, v6, v8], [v0, v1, v10, v2, v3, v5, v6, v8], [v0, v10, v2, v4, v5, v6, v8], [v1, v10, v2, v4, v5, v6, v8], [v0, v1, v10, v2, v4, v5, v6, v8], [v1, v10, v3, v4, v5, v6, v8], [v0, v1, v10, v3, v4, v5, v6, v8], [v0, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v10, v3, v5, v7, v8], [v0, v10, v2, v3, v5, v7, v8], [v1, v10, v2, v3, v5, v7, v8], [v0, v1, v10, v2, v3, v5, v7, v8], [v0, v10, v2, v4, v5, v7, v8], [v1, v10, v2, v4, v5, v7, v8], [v0, v1, v10, v2, v4, v5, v7, v8], [v1, v10, v3, v4, v5, v7, v8], [v0, v1, v10, v3, v4, v5, v7, v8], [v0, v10, v2, v3, v4, v5, v7, v8], [v1, v10, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v7, v8], [v0, v10, v2, v4, v6, v7, v8], [v1, v10, v2, v4, v6, v7, v8], [v0, v1, v10, v2, v4, v6, v7, v8], [v1, v10, v3, v4, v6, v7, v8], [v0, v1, v10, v3, v4, v6, v7, v8], [v0, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v3, v5, v6, v7, v8], [v0, v1, v10, v3, v5, v6, v7, v8], [v0, v10, v2, v3, v5, v6, v7, v8], [v1, v10, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v5, v6, v7, v8], [v0, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v4, v5, v6, v7, v8], [v0, v10, v2, v3, v4, v5, v6, v7, v8], [v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v10, v2, v3, v5, v7, v9], [v1, v10, v2, v3, v5, v7, v9], [v0, v1, v10, v2, v3, v5, 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v7, v8, v9], [v1, v10, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v2, v4, v6, v8, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v1, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v0, v1, v10, v3, v5, v6, v8], [v0, v10, v2, v3, v5, v6, v8], [v1, v10, v2, v3, v5, v6, v8], [v0, v1, v10, v2, v3, v5, v6, v8], [v0, v10, v2, v4, v5, v6, v8], [v1, v10, v2, v4, v5, v6, v8], [v0, v1, v10, v2, v4, v5, v6, v8], [v1, v10, v3, v4, v5, v6, v8], [v0, v1, v10, v3, v4, v5, v6, v8], [v0, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v2, v3, v4, v5, v6, v8], [v0, v1, 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v10, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v0, v10, v2, v4, v6, v8, v9], [v1, v10, v2, v4, v6, v8, v9], [v0, v1, v10, v2, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v1, v10, v3, v4, v6, v8, v9], [v0, v1, v10, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v0, v10, v2, v3, v4, v6, v8, v9], [v1, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v1, v10, v3, v5, v6, v8, v9], [v0, v1, v10, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v10, v2, v3, v5, v6, v8, v9], [v1, v10, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v0, v10, v2, v4, v5, v6, v8, v9], [v1, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v1, v10, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v1, v10, v3, v5, v7, v8, v9], [v0, v1, v10, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v10, v2, v3, v5, v7, v8, v9], [v1, v10, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v0, v10, v2, v4, v5, v7, v8, v9], [v1, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v1, v10, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v0, v10, v2, v4, v6, v7, v8, v9], [v1, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v1, v10, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v1, v10, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v2, v4, v6, v8, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156017,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v10\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0018","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v11\"],[\"v11\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v0, v10, v11, v2, v4, v6, v8], [v1, v10, v11, v2, v4, v6, v8], [v0, v1, v10, v11, v2, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v1, v10, v11, v3, v4, v6, v8], [v0, v1, v10, v11, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v0, v10, v11, v2, v3, v4, v6, v8], [v1, v10, v11, v2, v3, v4, v6, v8], [v0, v1, v10, v11, v2, v3, v4, v6, v8], [v0, v1, v10, v3, v5, v6, v8], [v1, v10, v11, v3, v5, v6, v8], [v0, v1, v10, v11, v3, v5, v6, v8], [v0, v10, v2, v3, v5, v6, v8], [v0, v1, v10, v2, v3, v5, v6, v8], [v0, v10, v11, v2, v3, v5, v6, v8], [v1, v10, v11, v2, v3, v5, v6, v8], [v0, v1, v10, v11, v2, v3, v5, v6, v8], [v0, v10, v2, v4, v5, v6, v8], [v0, v1, v10, v2, v4, v5, v6, v8], [v0, v10, v11, v2, v4, v5, v6, v8], [v1, v10, v11, v2, v4, v5, v6, v8], [v0, v1, v10, v11, v2, v4, v5, v6, v8], [v0, v1, v10, v3, v4, v5, v6, v8], [v1, v10, v11, v3, v4, v5, v6, v8], [v0, v1, v10, v11, v3, v4, v5, v6, v8], [v0, v10, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v8], [v0, v10, v11, v2, v3, v4, v5, v6, v8], [v1, v10, v11, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v3, v5, v7, v8], [v1, v10, v11, v3, v5, v7, v8], [v0, v1, v10, v11, v3, v5, v7, v8], [v0, v10, v2, v3, v5, v7, v8], [v0, v1, v10, v2, v3, v5, v7, v8], [v0, v10, v11, v2, v3, v5, v7, v8], [v1, v10, v11, v2, v3, v5, v7, v8], [v0, v1, v10, v11, v2, v3, v5, v7, v8], [v0, v10, v2, v4, v5, v7, v8], [v0, v1, v10, v2, v4, v5, v7, v8], [v0, v10, v11, v2, v4, v5, v7, v8], [v1, v10, v11, v2, v4, v5, v7, v8], [v0, v1, v10, v11, v2, v4, v5, v7, v8], [v0, v1, v10, v3, v4, v5, v7, v8], [v1, v10, v11, v3, v4, v5, v7, v8], [v0, v1, v10, v11, v3, v4, v5, v7, v8], [v0, v10, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v7, v8], [v0, v10, v11, v2, v3, v4, v5, v7, v8], [v1, v10, v11, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v11, v2, v3, v4, v5, v7, v8], [v0, v10, v2, v4, v6, v7, v8], [v0, v1, v10, v2, v4, v6, v7, v8], [v0, v10, v11, v2, v4, v6, v7, v8], [v1, v10, v11, v2, v4, v6, v7, v8], [v0, v1, v10, v11, v2, v4, v6, v7, v8], [v0, v1, v10, v3, v4, v6, v7, v8], [v1, v10, v11, v3, v4, v6, v7, v8], [v0, v1, v10, v11, v3, v4, v6, v7, v8], [v0, v10, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v6, v7, v8], [v0, v10, v11, v2, v3, v4, v6, v7, v8], [v1, v10, v11, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v3, v5, v6, v7, v8], [v1, v10, v11, v3, v5, v6, v7, v8], [v0, v1, v10, v11, v3, v5, v6, v7, v8], [v0, v10, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v5, v6, v7, v8], [v0, v10, v11, v2, v3, v5, v6, v7, v8], [v1, v10, v11, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v11, v2, v3, v5, v6, v7, v8], [v0, v10, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v4, v5, v6, v7, v8], [v0, v10, v11, v2, v4, v5, v6, v7, v8], [v1, v10, v11, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v11, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v4, v5, v6, v7, v8], [v1, v10, v11, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v11, v3, v4, v5, v6, v7, v8], [v0, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v8], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v5, v7, v9], [v1, v11, v3, v5, v7, v9], [v0, v1, v11, v3, v5, v7, v9], [v1, v10, v11, v3, v5, v7, v9], [v0, v1, v10, v11, v3, v5, v7, v9], [v0, v10, v2, v3, v5, v7, v9], [v0, v1, v10, v2, v3, v5, v7, v9], [v0, v11, v2, v3, v5, v7, v9], [v1, v11, v2, v3, v5, v7, v9], [v0, v1, v11, v2, v3, v5, v7, v9], [v0, v10, v11, v2, v3, v5, v7, v9], [v1, v10, v11, v2, v3, v5, v7, v9], [v0, v1, v10, v11, v2, v3, v5, v7, v9], [v0, v10, v2, v4, v5, v7, v9], [v0, v1, v10, v2, v4, v5, v7, v9], [v0, v11, v2, v4, v5, v7, v9], [v1, v11, v2, v4, v5, v7, v9], [v0, v1, v11, v2, v4, v5, v7, v9], [v0, v10, v11, v2, v4, v5, v7, v9], [v1, v10, v11, v2, v4, v5, v7, v9], [v0, v1, v10, v11, v2, v4, v5, v7, v9], [v0, v1, v10, v3, v4, 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v2, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v11, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v0, v10, v11, v2, v4, v6, v8], [v1, v10, v11, v2, v4, v6, v8], [v0, v1, v10, v11, v2, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v1, v10, v11, v3, v4, v6, v8], [v0, v1, v10, v11, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v0, v10, v11, v2, v3, v4, v6, v8], [v1, v10, v11, v2, v3, v4, v6, v8], 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v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v11, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156018,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v11"],["v11","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v11\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0019","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3], [[v0, v1], [v0, v2], [v0, v3]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3], [[v0, v1], [v0, v2], [v0, v3]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156019,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0020","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156020,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0021","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156021,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0022","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156022,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0023","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156023,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0024","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v8], [v0, v1, v8], [v0, v2, v8], [v0, v1, v2, v8], [v0, v3, v8], [v0, v1, v3, v8], [v0, v2, v3, v8], [v0, v1, v2, v3, v8], [v0, v4, v8], [v0, v1, v4, v8], [v0, v2, v4, v8], [v0, v1, v2, v4, v8], [v0, v3, v4, v8], [v0, v1, v3, v4, v8], [v0, v2, v3, v4, v8], [v0, v1, v2, v3, v4, v8], [v0, v5, v8], [v0, v1, v5, v8], [v0, v2, v5, v8], [v0, v1, v2, v5, v8], [v0, v3, v5, v8], [v0, v1, v3, v5, v8], [v0, v2, v3, v5, v8], [v0, v1, v2, v3, v5, v8], [v0, v4, v5, v8], [v0, v1, v4, v5, v8], [v0, v2, v4, v5, v8], [v0, v1, v2, v4, v5, v8], [v0, v3, v4, v5, v8], [v0, v1, v3, v4, v5, v8], [v0, v2, v3, v4, v5, v8], [v0, v1, v2, v3, v4, v5, v8], [v0, v6, v8], [v0, v1, v6, v8], [v0, v2, v6, v8], [v0, v1, v2, v6, v8], [v0, v3, v6, v8], [v0, v1, v3, v6, v8], [v0, v2, v3, v6, v8], [v0, v1, v2, v3, v6, v8], [v0, v4, v6, v8], [v0, v1, v4, v6, v8], [v0, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v0, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v0, v5, v6, v8], [v0, v1, v5, v6, v8], [v0, v2, v5, v6, v8], [v0, v1, v2, v5, v6, v8], [v0, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v4, v5, v6, v8], [v0, v1, v4, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v0, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v0, v7, v8], [v0, v1, v7, v8], [v0, v2, v7, v8], [v0, v1, v2, v7, v8], [v0, v3, v7, v8], [v0, v1, v3, v7, v8], [v0, v2, v3, v7, v8], [v0, v1, v2, v3, v7, v8], [v0, v4, v7, v8], [v0, v1, v4, v7, v8], [v0, v2, v4, v7, v8], [v0, v1, v2, v4, v7, v8], [v0, v3, v4, v7, v8], [v0, v1, v3, v4, v7, v8], [v0, v2, v3, v4, v7, v8], [v0, v1, v2, v3, v4, v7, v8], [v0, v5, v7, v8], [v0, v1, v5, v7, v8], [v0, v2, v5, v7, v8], [v0, v1, v2, v5, v7, v8], [v0, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v4, v5, v7, v8], [v0, v1, v4, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v0, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v6, v7, v8], [v0, v1, v6, v7, v8], [v0, v2, v6, v7, v8], [v0, v1, v2, v6, v7, v8], [v0, v3, v6, v7, v8], [v0, v1, v3, v6, v7, v8], [v0, v2, v3, v6, v7, v8], [v0, v1, v2, v3, v6, v7, v8], [v0, v4, v6, v7, v8], [v0, v1, v4, v6, v7, v8], [v0, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v0, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v0, v5, v6, v7, v8], [v0, v1, v5, v6, v7, v8], [v0, v2, v5, v6, v7, v8], [v0, v1, v2, v5, v6, v7, v8], [v0, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v4, v5, v6, v7, v8], [v0, v1, v4, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v0, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v8], [v0, v1, v8], [v0, v2, v8], [v0, v1, v2, v8], [v0, v3, v8], [v0, v1, v3, v8], [v0, v2, v3, v8], [v0, v1, v2, v3, v8], [v0, v4, v8], [v0, v1, v4, v8], [v0, v2, v4, v8], [v0, v1, v2, v4, v8], [v0, v3, v4, v8], [v0, v1, v3, v4, v8], [v0, v2, v3, v4, v8], [v0, v1, v2, v3, v4, v8], [v0, v5, v8], [v0, v1, v5, v8], [v0, v2, v5, v8], [v0, v1, v2, v5, v8], [v0, v3, v5, v8], [v0, v1, v3, v5, v8], [v0, v2, v3, v5, v8], [v0, v1, v2, v3, v5, v8], [v0, v4, v5, v8], [v0, v1, v4, v5, v8], [v0, v2, v4, v5, v8], [v0, v1, v2, v4, v5, v8], [v0, v3, v4, v5, v8], [v0, v1, v3, v4, v5, v8], [v0, v2, v3, v4, v5, v8], [v0, v1, v2, v3, v4, v5, v8], [v0, v6, v8], [v0, v1, v6, v8], [v0, v2, v6, v8], [v0, v1, v2, v6, v8], [v0, v3, v6, v8], [v0, v1, v3, v6, v8], [v0, v2, v3, v6, v8], [v0, v1, v2, v3, v6, v8], [v0, v4, v6, v8], [v0, v1, v4, v6, v8], [v0, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v0, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v0, v5, v6, v8], [v0, v1, v5, v6, v8], [v0, v2, v5, v6, v8], [v0, v1, v2, v5, v6, v8], [v0, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v4, v5, v6, v8], [v0, v1, v4, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v0, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v0, v7, v8], [v0, v1, v7, v8], [v0, v2, v7, v8], [v0, v1, v2, v7, v8], [v0, v3, v7, v8], [v0, v1, v3, v7, v8], [v0, v2, v3, v7, v8], [v0, v1, v2, v3, v7, v8], [v0, v4, v7, v8], [v0, v1, v4, v7, v8], [v0, v2, v4, v7, v8], [v0, v1, v2, v4, v7, v8], [v0, v3, v4, v7, v8], [v0, v1, v3, v4, v7, v8], [v0, v2, v3, v4, v7, v8], [v0, v1, v2, v3, v4, v7, v8], [v0, v5, v7, v8], [v0, v1, v5, v7, v8], [v0, v2, v5, v7, v8], [v0, v1, v2, v5, v7, v8], [v0, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v4, v5, v7, v8], [v0, v1, v4, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v0, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v6, v7, v8], [v0, v1, v6, v7, v8], [v0, v2, v6, v7, v8], [v0, v1, v2, v6, v7, v8], [v0, v3, v6, v7, v8], [v0, v1, v3, v6, v7, v8], [v0, v2, v3, v6, v7, v8], [v0, v1, v2, v3, v6, v7, v8], [v0, v4, v6, v7, v8], [v0, v1, v4, v6, v7, v8], [v0, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v0, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v0, v5, v6, v7, v8], [v0, v1, v5, v6, v7, v8], [v0, v2, v5, v6, v7, v8], [v0, v1, v2, v5, v6, v7, v8], [v0, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v4, v5, v6, v7, v8], [v0, v1, v4, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v0, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156024,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"],["v0","v8"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0025","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-exhaustive-minimizers；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, 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v9], [v0, v1, v4, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v0, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v0, v7, v8, v9], [v0, v1, v7, v8, v9], [v0, v2, v7, v8, v9], [v0, v1, v2, v7, v8, v9], [v0, v3, v7, v8, v9], [v0, v1, v3, v7, v8, v9], [v0, v2, v3, v7, v8, v9], [v0, v1, v2, v3, v7, v8, v9], [v0, v4, v7, v8, v9], [v0, v1, v4, v7, v8, v9], [v0, v2, v4, v7, v8, v9], [v0, v1, v2, v4, v7, v8, v9], [v0, v3, v4, v7, v8, v9], [v0, v1, v3, v4, v7, v8, v9], [v0, v2, v3, v4, v7, v8, v9], [v0, v1, v2, v3, v4, v7, v8, v9], [v0, v5, v7, v8, v9], [v0, v1, v5, v7, v8, v9], [v0, v2, v5, v7, v8, v9], [v0, v1, v2, v5, v7, v8, v9], [v0, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v4, v5, v7, v8, v9], [v0, v1, v4, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v0, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v6, v7, v8, v9], [v0, v1, v6, v7, v8, v9], [v0, v2, v6, v7, v8, v9], [v0, v1, v2, v6, v7, v8, v9], [v0, v3, v6, v7, v8, v9], [v0, v1, v3, v6, v7, v8, v9], [v0, v2, v3, v6, v7, v8, v9], [v0, v1, v2, v3, v6, v7, v8, v9], [v0, v4, v6, v7, v8, v9], [v0, v1, v4, v6, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v0, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v0, v5, v6, v7, v8, v9], [v0, v1, v5, v6, v7, v8, v9], [v0, v2, v5, v6, v7, v8, v9], [v0, v1, v2, v5, v6, v7, v8, v9], [v0, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v4, v5, v6, v7, v8, v9], [v0, v1, v4, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v0, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v8], [v0, v1, v8], [v0, v2, v8], [v0, v1, v2, v8], [v0, v3, v8], [v0, v1, v3, v8], [v0, v2, v3, v8], [v0, v1, v2, v3, v8], [v0, v4, v8], [v0, v1, v4, v8], [v0, v2, v4, v8], [v0, v1, v2, v4, v8], [v0, v3, v4, v8], [v0, v1, v3, v4, v8], [v0, v2, v3, v4, v8], [v0, v1, v2, v3, v4, v8], [v0, v5, v8], [v0, v1, v5, v8], [v0, v2, v5, v8], [v0, v1, v2, v5, v8], [v0, v3, v5, v8], [v0, v1, v3, v5, v8], [v0, v2, v3, v5, v8], [v0, v1, v2, v3, v5, v8], [v0, v4, v5, v8], [v0, v1, v4, v5, v8], [v0, v2, v4, v5, v8], [v0, v1, v2, v4, v5, v8], [v0, v3, v4, v5, v8], [v0, v1, v3, v4, v5, v8], [v0, v2, v3, v4, v5, v8], [v0, v1, v2, v3, v4, v5, v8], [v0, v6, v8], [v0, v1, v6, v8], [v0, v2, v6, v8], [v0, v1, v2, v6, v8], [v0, v3, v6, v8], [v0, v1, v3, v6, v8], [v0, v2, v3, v6, v8], [v0, v1, v2, v3, v6, v8], [v0, v4, v6, v8], [v0, v1, v4, v6, v8], [v0, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v0, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v0, v5, v6, v8], [v0, v1, v5, v6, v8], [v0, v2, v5, v6, v8], [v0, v1, v2, v5, v6, v8], [v0, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v4, v5, v6, v8], [v0, v1, v4, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v0, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v0, v7, v8], [v0, v1, v7, v8], [v0, v2, v7, v8], [v0, v1, v2, v7, v8], [v0, v3, v7, v8], [v0, v1, v3, v7, v8], [v0, v2, v3, v7, v8], [v0, v1, v2, v3, v7, v8], [v0, v4, v7, v8], [v0, v1, v4, v7, v8], [v0, v2, v4, v7, v8], [v0, v1, v2, v4, v7, v8], [v0, v3, v4, v7, v8], [v0, v1, v3, v4, v7, v8], [v0, v2, v3, v4, v7, v8], [v0, v1, v2, v3, v4, v7, v8], [v0, v5, v7, v8], [v0, v1, v5, v7, v8], [v0, v2, v5, v7, v8], [v0, v1, v2, v5, v7, v8], [v0, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v4, v5, v7, v8], [v0, v1, v4, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v0, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v6, v7, v8], [v0, v1, v6, v7, v8], [v0, v2, v6, v7, v8], [v0, v1, v2, v6, v7, v8], [v0, v3, v6, v7, v8], [v0, v1, v3, v6, v7, v8], [v0, v2, v3, v6, v7, v8], [v0, v1, v2, v3, v6, v7, v8], [v0, v4, v6, v7, v8], [v0, v1, v4, v6, v7, v8], [v0, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v0, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v0, v5, v6, v7, v8], [v0, v1, v5, v6, v7, v8], [v0, v2, v5, v6, v7, v8], [v0, v1, v2, v5, v6, v7, v8], [v0, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v4, v5, v6, v7, v8], [v0, v1, v4, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v0, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8], [v0, v9], [v0, v1, v9], [v0, v2, v9], [v0, v1, v2, v9], [v0, v3, v9], [v0, v1, v3, v9], [v0, v2, v3, v9], [v0, v1, v2, v3, v9], [v0, v4, v9], [v0, v1, v4, v9], [v0, v2, v4, v9], [v0, v1, v2, v4, v9], [v0, v3, v4, v9], [v0, v1, v3, v4, v9], [v0, v2, v3, v4, v9], [v0, v1, v2, v3, v4, v9], [v0, v5, v9], [v0, v1, v5, v9], [v0, v2, v5, v9], [v0, v1, v2, v5, v9], [v0, v3, v5, v9], [v0, v1, v3, v5, v9], [v0, v2, v3, v5, v9], [v0, v1, v2, v3, v5, v9], [v0, v4, v5, v9], [v0, v1, v4, v5, v9], [v0, v2, v4, v5, v9], [v0, v1, v2, v4, v5, v9], [v0, v3, v4, v5, v9], [v0, v1, v3, v4, v5, v9], [v0, v2, v3, v4, v5, v9], [v0, v1, v2, v3, v4, v5, v9], [v0, v6, v9], [v0, v1, v6, v9], [v0, v2, v6, v9], [v0, v1, v2, v6, v9], [v0, v3, v6, v9], [v0, v1, v3, v6, v9], [v0, v2, v3, v6, v9], [v0, v1, v2, v3, v6, v9], [v0, v4, v6, v9], [v0, v1, v4, v6, v9], [v0, v2, v4, v6, v9], [v0, v1, v2, v4, v6, v9], [v0, v3, v4, v6, v9], [v0, v1, v3, v4, v6, v9], [v0, v2, v3, v4, v6, v9], [v0, v1, v2, v3, v4, v6, v9], [v0, v5, v6, v9], [v0, v1, v5, v6, v9], [v0, v2, v5, v6, v9], [v0, v1, v2, v5, v6, v9], [v0, v3, v5, v6, v9], [v0, v1, v3, v5, v6, v9], [v0, v2, v3, v5, v6, v9], [v0, v1, v2, v3, v5, v6, v9], [v0, v4, v5, v6, v9], [v0, v1, v4, v5, v6, v9], [v0, v2, v4, v5, v6, v9], [v0, v1, v2, v4, v5, v6, v9], [v0, v3, v4, v5, v6, v9], [v0, v1, v3, v4, v5, v6, v9], [v0, v2, v3, v4, v5, v6, v9], [v0, v1, v2, v3, v4, v5, v6, v9], [v0, v7, v9], [v0, v1, v7, v9], [v0, v2, v7, v9], [v0, v1, v2, v7, v9], [v0, v3, v7, v9], [v0, v1, v3, v7, v9], [v0, v2, v3, v7, v9], [v0, v1, v2, v3, v7, v9], [v0, v4, v7, v9], [v0, v1, v4, v7, v9], [v0, v2, v4, v7, v9], [v0, v1, v2, v4, v7, v9], [v0, v3, v4, v7, v9], [v0, v1, v3, v4, v7, v9], [v0, v2, v3, v4, v7, v9], [v0, v1, v2, v3, v4, v7, v9], [v0, v5, v7, v9], [v0, v1, v5, v7, v9], [v0, v2, v5, v7, v9], [v0, v1, v2, v5, v7, v9], [v0, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v4, v5, v7, v9], [v0, v1, v4, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v0, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v6, v7, v9], [v0, v1, v6, v7, v9], [v0, v2, v6, v7, v9], [v0, v1, v2, v6, v7, v9], [v0, v3, v6, v7, v9], [v0, v1, v3, v6, v7, v9], [v0, v2, v3, v6, v7, v9], [v0, v1, v2, v3, v6, v7, v9], [v0, v4, v6, v7, v9], [v0, v1, v4, v6, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v0, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v0, v5, v6, v7, v9], [v0, v1, v5, v6, v7, v9], [v0, v2, v5, v6, v7, v9], [v0, v1, v2, v5, v6, v7, v9], [v0, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v4, v5, v6, v7, v9], [v0, v1, v4, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v0, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v8, v9], [v0, v1, v8, v9], [v0, v2, v8, v9], [v0, v1, v2, v8, v9], [v0, v3, v8, v9], [v0, v1, v3, v8, v9], [v0, v2, v3, v8, v9], [v0, v1, v2, v3, v8, v9], [v0, v4, v8, v9], [v0, v1, v4, v8, v9], [v0, v2, v4, v8, v9], [v0, v1, v2, v4, v8, v9], [v0, v3, v4, v8, v9], [v0, v1, v3, v4, v8, v9], [v0, v2, v3, v4, v8, v9], [v0, v1, v2, v3, v4, v8, v9], [v0, v5, v8, v9], [v0, v1, v5, v8, v9], [v0, v2, v5, v8, v9], [v0, v1, v2, v5, v8, v9], [v0, v3, v5, v8, v9], [v0, v1, v3, v5, v8, v9], [v0, v2, v3, v5, v8, v9], [v0, v1, v2, v3, v5, v8, v9], [v0, v4, v5, v8, v9], [v0, v1, v4, v5, v8, v9], [v0, v2, v4, v5, v8, v9], [v0, v1, v2, v4, v5, v8, v9], [v0, v3, v4, v5, v8, v9], [v0, v1, v3, v4, v5, v8, v9], [v0, v2, v3, v4, v5, v8, v9], [v0, v1, v2, v3, v4, v5, v8, v9], [v0, v6, v8, v9], [v0, v1, v6, v8, v9], [v0, v2, v6, v8, v9], [v0, v1, v2, v6, v8, v9], [v0, v3, v6, v8, v9], [v0, v1, v3, v6, v8, v9], [v0, v2, v3, v6, v8, v9], [v0, v1, v2, v3, v6, v8, v9], [v0, v4, v6, v8, v9], [v0, v1, v4, v6, v8, v9], [v0, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v0, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v0, v5, v6, v8, v9], [v0, v1, v5, v6, v8, v9], [v0, v2, v5, v6, v8, v9], [v0, v1, v2, v5, v6, v8, v9], [v0, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v4, v5, v6, v8, v9], [v0, v1, v4, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v0, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v0, v7, v8, v9], [v0, v1, v7, v8, v9], [v0, v2, v7, v8, v9], [v0, v1, v2, v7, v8, v9], [v0, v3, v7, v8, v9], [v0, v1, v3, v7, v8, v9], [v0, v2, v3, v7, v8, v9], [v0, v1, v2, v3, v7, v8, v9], [v0, v4, v7, v8, v9], [v0, v1, v4, v7, v8, v9], [v0, v2, v4, v7, v8, v9], [v0, v1, v2, v4, v7, v8, v9], [v0, v3, v4, v7, v8, v9], [v0, v1, v3, v4, v7, v8, v9], [v0, v2, v3, v4, v7, v8, v9], [v0, v1, v2, v3, v4, v7, v8, v9], [v0, v5, v7, v8, v9], [v0, v1, v5, v7, v8, v9], [v0, v2, v5, v7, v8, v9], [v0, v1, v2, v5, v7, v8, v9], [v0, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v4, v5, v7, v8, v9], [v0, v1, v4, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v0, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v6, v7, v8, v9], [v0, v1, v6, v7, v8, v9], [v0, v2, v6, v7, v8, v9], [v0, v1, v2, v6, v7, v8, v9], [v0, v3, v6, v7, v8, v9], [v0, v1, v3, v6, v7, v8, v9], [v0, v2, v3, v6, v7, v8, v9], [v0, v1, v2, v3, v6, v7, v8, v9], [v0, v4, v6, v7, v8, v9], [v0, v1, v4, v6, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v0, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v0, v5, v6, v7, v8, v9], [v0, v1, v5, v6, v7, v8, v9], [v0, v2, v5, v6, v7, v8, v9], [v0, v1, v2, v5, v6, v7, v8, v9], [v0, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v4, v5, v6, v7, v8, v9], [v0, v1, v4, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v0, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156025,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"],["v0","v8"],["v0","v9"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-exhaustive-minimizers","semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0026","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3], 是, 4, 2, [v0, v2], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3], 是, 4, 2, [v0, v2], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156026,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"]],"candidate":["v0","v1","v2","v3"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0027","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], 是, 5, 2, [v1, v3], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4], 是, 5, 2, [v1, v3], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156027,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"]],"candidate":["v0","v1","v2","v3","v4"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0028","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], 是, 6, 3, [v0, v2, v4], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5], 是, 6, 3, [v0, v2, v4], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156028,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"]],"candidate":["v0","v1","v2","v3","v4","v5"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0029","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], 是, 7, 3, [v1, v3, v5], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6], 是, 7, 3, [v1, v3, v5], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156029,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"]],"candidate":["v0","v1","v2","v3","v4","v5","v6"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0030","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], 是, 8, 4, [v0, v2, v4, v6], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7], 是, 8, 4, [v0, v2, v4, v6], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156030,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0031","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, 9, 4, [v1, v3, v5, v7], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, 9, 4, [v1, v3, v5, v7], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156031,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0032","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, 10, 5, [v0, v2, v4, v6, v8], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, 10, 5, [v0, v2, v4, v6, v8], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156032,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0033","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, 11, 5, [v1, v3, v5, v7, v9], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, 11, 5, [v1, v3, v5, v7, v9], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156033,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0034","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v11\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], 是, 12, 6, [v0, v10, v2, v4, v6, v8], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], 是, 12, 6, [v0, v10, v2, v4, v6, v8], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156034,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v11"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0035","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3], 是, 4, 2, [v0, v2], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3], 是, 4, 2, [v0, v2], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156035,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v0"]],"candidate":["v0","v1","v2","v3"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0036","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], 是, 5, 3, [v0, v1, v3], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4], 是, 5, 3, [v0, v1, v3], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156036,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v0"]],"candidate":["v0","v1","v2","v3","v4"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0037","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], 是, 6, 3, [v0, v2, v4], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5], 是, 6, 3, [v0, v2, v4], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156037,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v0"]],"candidate":["v0","v1","v2","v3","v4","v5"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0038","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], 是, 7, 4, [v0, v1, v3, v5], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6], 是, 7, 4, [v0, v1, v3, v5], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156038,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v0"]],"candidate":["v0","v1","v2","v3","v4","v5","v6"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v6\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0039","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], 是, 8, 4, [v0, v2, v4, v6], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7], 是, 8, 4, [v0, v2, v4, v6], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156039,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v0"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v7\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0040","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, 9, 5, [v0, v1, v3, v5, v7], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, 9, 5, [v0, v1, v3, v5, v7], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156040,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v0"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v8\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0041","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, 10, 5, [v0, v2, v4, v6, v8], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, 10, 5, [v0, v2, v4, v6, v8], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156041,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v0"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v9\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0042","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, 11, 6, [v0, v10, v2, v4, v6, v8], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, 11, 6, [v0, v10, v2, v4, v6, v8], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156042,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v0"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v10\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0043","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v11\"],[\"v11\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], 是, 12, 6, [v0, v10, v2, v4, v6, v8], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], 是, 12, 6, [v0, v10, v2, v4, v6, v8], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156043,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v11"],["v11","v0"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v11\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0044","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3], 是, 4, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3], 是, 4, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156044,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"]],"candidate":["v0","v1","v2","v3"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0045","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], 是, 5, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4], 是, 5, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156045,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"]],"candidate":["v0","v1","v2","v3","v4"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0046","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], 是, 6, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5], 是, 6, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156046,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"]],"candidate":["v0","v1","v2","v3","v4","v5"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0047","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], 是, 7, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6], 是, 7, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156047,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"]],"candidate":["v0","v1","v2","v3","v4","v5","v6"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0048","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], 是, 8, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7], 是, 8, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156048,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0049","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, 9, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, 9, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156049,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"],["v0","v8"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0050","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, 10, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)","explanation_zh":"Wave6第九層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, 10, 1, [v0], 否, explicit_smaller_cover_refutes_minimum_status)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156050,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"],["v0","v8"],["v0","v9"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-nonminimum-witness","semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0051","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"]],\"candidate\":[\"l0_0\",\"l0_1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1], [[l0_0, [l0_1], 否], [l0_1, [l0_0], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1], [[l0_0, [l0_1], 否], [l0_1, [l0_0], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156051,"parameters":{"vertices":["c0","l0_0","l0_1"],"edges":[["c0","l0_0"],["c0","l0_1"]],"candidate":["l0_0","l0_1"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0052","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2], [[l0_0, [l0_1, l0_2], 否], [l0_1, [l0_0, l0_2], 否], [l0_2, [l0_0, l0_1], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2], [[l0_0, [l0_1, l0_2], 否], [l0_1, [l0_0, l0_2], 否], [l0_2, [l0_0, l0_1], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156052,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"]],"candidate":["l0_0","l0_1","l0_2"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0053","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3], [[l0_0, [l0_1, l0_2, l0_3], 否], [l0_1, [l0_0, l0_2, l0_3], 否], [l0_2, [l0_0, l0_1, l0_3], 否], [l0_3, [l0_0, l0_1, l0_2], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l0_3], [[l0_0, [l0_1, l0_2, l0_3], 否], [l0_1, [l0_0, l0_2, l0_3], 否], [l0_2, [l0_0, l0_1, l0_3], 否], [l0_3, [l0_0, l0_1, l0_2], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156053,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"]],"candidate":["l0_0","l0_1","l0_2","l0_3"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0054","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4], [[l0_0, [l0_1, l0_2, l0_3, l0_4], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l0_3, l0_4], [[l0_0, [l0_1, l0_2, l0_3, l0_4], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156054,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0055","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156055,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4","l0_5"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"],["c0","l0_5"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4","l0_5"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0056","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156056,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"],["c0","l0_5"],["c0","l0_6"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0057","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7], 否], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7], 否], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156057,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"],["c0","l0_5"],["c0","l0_6"],["c0","l0_7"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0058","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8], 否], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8], 否], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8], 否], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8], 否], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156058,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7","l0_8"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"],["c0","l0_5"],["c0","l0_6"],["c0","l0_7"],["c0","l0_8"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7","l0_8"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0059","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"],[\"c0\",\"l0_9\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], 否], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], 否], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], 否], [l0_9, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], 否], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], 否], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], 否], [l0_9, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156059,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7","l0_8","l0_9"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"],["c0","l0_5"],["c0","l0_6"],["c0","l0_7"],["c0","l0_8"],["c0","l0_9"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7","l0_8","l0_9"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"],[\"c0\",\"l0_9\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0060","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\",\"l0_10\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"],[\"c0\",\"l0_9\"],[\"c0\",\"l0_10\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\",\"l0_10\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_1, [l0_0, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_10, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_2, [l0_0, l0_1, l0_10, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_3, [l0_0, l0_1, l0_10, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_4, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_5, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], 否], [l0_6, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], 否], [l0_7, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], 否], [l0_8, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], 否], [l0_9, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_1, [l0_0, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_10, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_2, [l0_0, l0_1, l0_10, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_3, [l0_0, l0_1, l0_10, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_4, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_5, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], 否], [l0_6, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], 否], [l0_7, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], 否], [l0_8, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], 否], [l0_9, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否]], 是, 1, [c0], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156060,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7","l0_8","l0_9","l0_10"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"],["c0","l0_5"],["c0","l0_6"],["c0","l0_7"],["c0","l0_8"],["c0","l0_9"],["c0","l0_10"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7","l0_8","l0_9","l0_10"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_10\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_10\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"],[\"c0\",\"l0_9\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_10\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0061","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1], [[l0_0, [l0_1, l1_0, l1_1], 否], [l0_1, [l0_0, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l1_1], 否], [l1_1, [l0_0, l0_1, l1_0], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l1_0, l1_1], [[l0_0, [l0_1, l1_0, l1_1], 否], [l0_1, [l0_0, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l1_1], 否], [l1_1, [l0_0, l0_1, l1_0], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156061,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"]],"candidate":["l0_0","l0_1","l1_0","l1_1"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0062","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l1_0, l1_1, l1_2], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l1_0, l1_1, l1_2], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156062,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1","l1_2"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"]],"candidate":["l0_0","l0_1","l1_0","l1_1","l1_2"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0063","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156063,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1","l1_2","l1_3"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"]],"candidate":["l0_0","l0_1","l1_0","l1_1","l1_2","l1_3"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0064","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156064,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1","l1_2","l1_3","l1_4"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"]],"candidate":["l0_0","l0_1","l1_0","l1_1","l1_2","l1_3","l1_4"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0065","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156065,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"],["c1","l1_5"]],"candidate":["l0_0","l0_1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0066","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], 否], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], 否], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156066,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5","l1_6"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"],["c1","l1_5"],["c1","l1_6"]],"candidate":["l0_0","l0_1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5","l1_6"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0067","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\",\"l1_7\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"],[\"c1\",\"l1_7\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\",\"l1_7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6, l1_7], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6, l1_7], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6, l1_7], 否], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_7], 否], [l1_7, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6, l1_7], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6, l1_7], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6, l1_7], 否], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_7], 否], [l1_7, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156067,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5","l1_6","l1_7"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"],["c1","l1_5"],["c1","l1_6"],["c1","l1_7"]],"candidate":["l0_0","l0_1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5","l1_6","l1_7"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\",\"l1_7\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"],[\"c1\",\"l1_7\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\",\"l1_7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0068","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"c1\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l1_0, l1_1], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l1_0, l1_1], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156068,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0069","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156069,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1","l1_2"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1","l1_2"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0070","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156070,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1","l1_2","l1_3"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1","l1_2","l1_3"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0071","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], 否], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], 否], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156071,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1","l1_2","l1_3","l1_4"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1","l1_2","l1_3","l1_4"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0072","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5], 否], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5], 否], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5], 否], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5], 否], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156072,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"],["c1","l1_5"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0073","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], 否], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], 否], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], 否], [l1_6, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], 否], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], 否], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], 否], [l1_6, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156073,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5","l1_6"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"],["c1","l1_5"],["c1","l1_6"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5","l1_6"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0074","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"c1\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1], 否], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1], 否], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1], 否], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1], 否], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1], 否], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1], 否], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1], 否], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1], 否], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156074,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","c1","l1_0","l1_1"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c1","l1_0"],["c1","l1_1"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l1_0","l1_1"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0075","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\",\"l1_2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], 否], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1, l1_2], 否], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1, l1_2], 否], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], 否], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1, l1_2], 否], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_2], 否], [l1_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)","explanation_zh":"Wave6第九層核對=([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], 否], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1, l1_2], 否], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1, l1_2], 否], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], 否], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1, l1_2], 否], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_2], 否], [l1_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156075,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","c1","l1_0","l1_1","l1_2"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l1_0","l1_1","l1_2"],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-minimal-not-minimum","semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\",\"l1_2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0076","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"empty_graph\",\"vertices\":[\"v0\"],\"edges\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156076,"parameters":{"mode":"empty_graph","vertices":["v0"],"edges":[],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0077","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\"],\"edges\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156077,"parameters":{"mode":"empty_graph","vertices":["v0","v1"],"edges":[],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0078","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\"],\"edges\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156078,"parameters":{"mode":"empty_graph","vertices":["v0","v1","v2"],"edges":[],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0079","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156079,"parameters":{"mode":"empty_graph","vertices":["v0","v1","v2","v3"],"edges":[],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0080","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156080,"parameters":{"mode":"empty_graph","vertices":["v0","v1","v2","v3","v4"],"edges":[],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0081","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156081,"parameters":{"mode":"empty_graph","vertices":["v0","v1","v2","v3","v4","v5"],"edges":[],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0082","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156082,"parameters":{"mode":"empty_graph","vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0083","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156083,"parameters":{"mode":"empty_graph","vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0084","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156084,"parameters":{"mode":"empty_graph","vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0085","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156085,"parameters":{"mode":"empty_graph","vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0086","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"edges\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156086,"parameters":{"mode":"empty_graph","vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0087","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"],\"edges\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156087,"parameters":{"mode":"empty_graph","vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0088","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v2], [v1, v2]], 2, [[v0, v1], [v0, v2], [v1, v2]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(multiple_minima, [[v0, v1], [v0, v2], [v1, v2]], 2, [[v0, v1], [v0, v2], [v1, v2]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156088,"parameters":{"mode":"multiple_minima","vertices":["v0","v1","v2"],"edges":[["v0","v1"],["v1","v2"],["v2","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v1\",\"v2\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0089","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], 2, [[v0, v2], [v1, v3]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(multiple_minima, [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], 2, [[v0, v2], [v1, v3]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156089,"parameters":{"mode":"multiple_minima","vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0090","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], 3, [[v0, v1, v3], [v0, v2, v3], [v0, v2, v4], [v1, v2, v4], [v1, v3, v4]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(multiple_minima, [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], 3, [[v0, v1, v3], [v0, v2, v3], [v0, v2, v4], [v1, v2, v4], [v1, v3, v4]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156090,"parameters":{"mode":"multiple_minima","vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0091","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], 3, [[v0, v2, v4], [v1, v3, v5]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(multiple_minima, [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], 3, [[v0, v2, v4], [v1, v3, v5]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156091,"parameters":{"mode":"multiple_minima","vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0092","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], 4, [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v2, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(multiple_minima, [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], 4, [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v2, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156092,"parameters":{"mode":"multiple_minima","vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v6\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0093","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], 4, [[v0, v2, v4, v6], [v1, v3, v5, v7]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(multiple_minima, [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], 4, [[v0, v2, v4, v6], [v1, v3, v5, v7]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156093,"parameters":{"mode":"multiple_minima","vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v7\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0094","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(multiple_minima, [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156094,"parameters":{"mode":"multiple_minima","vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v8\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0095","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 5, [[v0, v2, v4, v6, v8], [v1, v3, v5, v7, v9]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(multiple_minima, [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 5, [[v0, v2, v4, v6, v8], [v1, v3, v5, v7, v9]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156095,"parameters":{"mode":"multiple_minima","vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v9\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0096","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v2, v4, v6, v8, v9]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(multiple_minima, [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v2, v4, v6, v8, v9]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156096,"parameters":{"mode":"multiple_minima","vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v10\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0097","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v11\"],[\"v11\",\"v0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v11, v3, v5, v7, v9]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(multiple_minima, [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v11, v3, v5, v7, v9]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156097,"parameters":{"mode":"multiple_minima","vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v11"],["v11","v0"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v11\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0098","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v1\",\"v3\"],[\"v2\",\"v3\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v2], [v0, v3], [v1, v2], [v1, v3], [v2, v3]], 3, [[v0, v1, v2], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(multiple_minima, [[v0, v1], [v0, v2], [v0, v3], [v1, v2], [v1, v3], [v2, v3]], 3, [[v0, v1, v2], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156098,"parameters":{"mode":"multiple_minima","vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v1","v2"],["v1","v3"],["v2","v3"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v1\",\"v3\"],[\"v2\",\"v3\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0099","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v1\",\"v3\"],[\"v1\",\"v4\"],[\"v2\",\"v3\"],[\"v2\",\"v4\"],[\"v3\",\"v4\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v1, v4], [v2, v3], [v2, v4], [v3, v4]], 4, [[v0, v1, v2, v3], [v0, v1, v2, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(multiple_minima, [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v1, v4], [v2, v3], [v2, v4], [v3, v4]], 4, [[v0, v1, v2, v3], [v0, v1, v2, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156099,"parameters":{"mode":"multiple_minima","vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v1","v2"],["v1","v3"],["v1","v4"],["v2","v3"],["v2","v4"],["v3","v4"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v1\",\"v3\"],[\"v1\",\"v4\"],[\"v2\",\"v3\"],[\"v2\",\"v4\"],[\"v3\",\"v4\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0100","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-minimum-vertex-cover"],"title_zh":"最小頂點覆蓋","stem_zh":"bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v1\",\"v3\"],[\"v1\",\"v4\"],[\"v1\",\"v5\"],[\"v2\",\"v3\"],[\"v2\",\"v4\"],[\"v2\",\"v5\"],[\"v3\",\"v4\"],[\"v3\",\"v5\"],[\"v4\",\"v5\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v1, v2], [v1, v3], [v1, v4], [v1, v5], [v2, v3], [v2, v4], [v2, v5], [v3, v4], [v3, v5], [v4, v5]], 5, [[v0, v1, v2, v3, v4], [v0, v1, v2, v3, v5], [v0, v1, v2, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation_zh":"Wave6第九層核對=(multiple_minima, [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v1, v2], [v1, v3], [v1, v4], [v1, v5], [v2, v3], [v2, v4], [v2, v5], [v3, v4], [v3, v5], [v4, v5]], 5, [[v0, v1, v2, v3, v4], [v0, v1, v2, v3, v5], [v0, v1, v2, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)；依完整明示資料獨立重算；minimum cardinality與全部minimizers由完整枚舉取得；nonminimum需更小cover witness；inclusion-minimal不是minimum；保留empty/multiple minima且不混同edge cover、independent set、König或weighted/directed variants。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最小頂點覆蓋"]},"variant":{"family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156100,"parameters":{"mode":"multiple_minima","vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v1","v2"],["v1","v3"],["v1","v4"],["v1","v5"],["v2","v3"],["v2","v4"],["v2","v5"],["v3","v4"],["v3","v5"],["v4","v5"]],"domain":"Complete_finite_minimum_vertex_cover","operation":"mincover-empty-multiple-boundary","semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v1\",\"v3\"],[\"v1\",\"v4\"],[\"v1\",\"v5\"],[\"v2\",\"v3\"],[\"v2\",\"v4\"],[\"v2\",\"v5\"],[\"v3\",\"v4\"],[\"v3\",\"v5\"],[\"v4\",\"v5\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}