{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0001","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[2,2,3,3,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, s, t], s, t, [[[s], [[[s, a0], 2], [[s, a1], 3]], 5], [[a0, s], [[[a0, t], 2], [[s, a1], 3]], 5], [[a1, s], [[[a1, t], 3], [[s, a0], 2]], 5], [[a0, a1, s], [[[a0, t], 2], [[a1, t], 3]], 5]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, s, t], s, t, [[[s], [[[s, a0], 2], [[s, a1], 3]], 5], [[a0, s], [[[a0, t], 2], [[s, a1], 3]], 5], [[a1, s], [[[a1, t], 3], [[s, a0], 2]], 5], [[a0, a1, s], [[[a0, t], 2], [[a1, t], 3]], 5]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155001,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[2,2,3,3,7],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[7,1],[2,1],[3,1],[2,1],[3,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0002","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[3,3,4,4,5,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a0, s], [[[a0, t], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a1, s], [[[a1, t], 4], [[s, a0], 3], [[s, a2], 5]], 12], [[a0, a1, s], [[[a0, t], 3], [[a1, t], 4], [[s, a2], 5]], 12], [[a2, s], [[[a2, t], 5], [[s, a0], 3], [[s, a1], 4]], 12], [[a0, a2, s], [[[a0, t], 3], [[a2, t], 5], [[s, a1], 4]], 12], [[a1, a2, s], [[[a1, t], 4], [[a2, t], 5], [[s, a0], 3]], 12], [[a0, a1, a2, s], [[[a0, t], 3], [[a1, t], 4], [[a2, t], 5]], 12]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a0, s], [[[a0, t], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a1, s], [[[a1, t], 4], [[s, a0], 3], [[s, a2], 5]], 12], [[a0, a1, s], [[[a0, t], 3], [[a1, t], 4], [[s, a2], 5]], 12], [[a2, s], [[[a2, t], 5], [[s, a0], 3], [[s, a1], 4]], 12], [[a0, a2, s], [[[a0, t], 3], [[a2, t], 5], [[s, a1], 4]], 12], [[a1, a2, s], [[[a1, t], 4], [[a2, t], 5], [[s, a0], 3]], 12], [[a0, a1, a2, s], [[[a0, t], 3], [[a1, t], 4], [[a2, t], 5]], 12]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155002,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[3,3,4,4,5,5],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[3,1],[4,1],[5,1],[3,1],[4,1],[5,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0003","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[4,4,5,5,6,6,7,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, s], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a1, s], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, a1, s], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a2, s], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22], [[a0, a2, s], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22], [[a1, a2, s], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22], [[a0, a1, a2, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22], [[a3, s], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22], [[a0, a3, s], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22], [[a1, a3, s], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22], [[a0, a1, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22], [[a2, a3, s], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22], [[a0, a2, a3, s], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22], [[a1, a2, a3, s], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22], [[a0, a1, a2, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, s], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a1, s], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, a1, s], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a2, s], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22], [[a0, a2, s], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22], [[a1, a2, s], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22], [[a0, a1, a2, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22], [[a3, s], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22], [[a0, a3, s], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22], [[a1, a3, s], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22], [[a0, a1, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22], [[a2, a3, s], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22], [[a0, a2, a3, s], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22], [[a1, a2, a3, s], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22], [[a0, a1, a2, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155003,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[4,4,5,5,6,6,7,7],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[4,1],[5,1],[6,1],[7,1],[4,1],[5,1],[6,1],[7,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0004","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[5,5,6,6,7,7,8,8,9,9,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, s], [[[a0, t], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, s], [[[a1, t], 6], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, s], [[[a0, t], 5], [[a1, t], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a2, s], [[[a2, t], 7], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a2, s], [[[a0, t], 5], [[a2, t], 7], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, a2, s], [[[a1, t], 6], [[a2, t], 7], [[s, a0], 5], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, a2, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a3, s], [[[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a3, s], [[[a0, t], 5], [[a3, t], 8], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a1, a3, s], [[[a1, t], 6], [[a3, t], 8], [[s, a0], 5], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a1, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[s, a2], 7], [[s, a4], 9]], 35], [[a2, a3, s], [[[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a4], 9]], 35], [[a0, a2, a3, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[s, a1], 6], [[s, a4], 9]], 35], [[a1, a2, a3, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a4], 9]], 35], [[a0, a1, a2, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a4], 9]], 35], [[a4, s], [[[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a4, s], [[[a0, t], 5], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a1, a4, s], [[[a1, t], 6], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a1, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a4, t], 9], [[s, a2], 7], [[s, a3], 8]], 35], [[a2, a4, s], [[[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8]], 35], [[a0, a2, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a4, t], 9], [[s, a1], 6], [[s, a3], 8]], 35], [[a1, a2, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a3], 8]], 35], [[a0, a1, a2, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a3], 8]], 35], [[a3, a4, s], [[[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7]], 35], [[a0, a3, a4, s], [[[a0, t], 5], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7]], 35], [[a1, a3, a4, s], [[[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7]], 35], [[a0, a1, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a2], 7]], 35], [[a2, a3, a4, s], [[[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6]], 35], [[a0, a2, a3, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6]], 35], [[a1, a2, a3, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5]], 35], [[a0, a1, a2, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9]], 35]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, s], [[[a0, t], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, s], [[[a1, t], 6], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, s], [[[a0, t], 5], [[a1, t], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a2, s], [[[a2, t], 7], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a2, s], [[[a0, t], 5], [[a2, t], 7], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, a2, s], [[[a1, t], 6], [[a2, t], 7], [[s, a0], 5], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, a2, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a3, s], [[[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a3, s], [[[a0, t], 5], [[a3, t], 8], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a1, a3, s], [[[a1, t], 6], [[a3, t], 8], [[s, a0], 5], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a1, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[s, a2], 7], [[s, a4], 9]], 35], [[a2, a3, s], [[[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a4], 9]], 35], [[a0, a2, a3, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[s, a1], 6], [[s, a4], 9]], 35], [[a1, a2, a3, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a4], 9]], 35], [[a0, a1, a2, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a4], 9]], 35], [[a4, s], [[[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a4, s], [[[a0, t], 5], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a1, a4, s], [[[a1, t], 6], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a1, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a4, t], 9], [[s, a2], 7], [[s, a3], 8]], 35], [[a2, a4, s], [[[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8]], 35], [[a0, a2, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a4, t], 9], [[s, a1], 6], [[s, a3], 8]], 35], [[a1, a2, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a3], 8]], 35], [[a0, a1, a2, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a3], 8]], 35], [[a3, a4, s], [[[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7]], 35], [[a0, a3, a4, s], [[[a0, t], 5], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7]], 35], [[a1, a3, a4, s], [[[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7]], 35], [[a0, a1, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a2], 7]], 35], [[a2, a3, a4, s], [[[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6]], 35], [[a0, a2, a3, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6]], 35], [[a1, a2, a3, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5]], 35], [[a0, a1, a2, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9]], 35]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155004,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"],["a0","s"]],"source":"s","sink":"t","capacities":[5,5,6,6,7,7,8,8,9,9,10],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[10,1],[5,1],[6,1],[7,1],[8,1],[9,1],[5,1],[6,1],[7,1],[8,1],[9,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0005","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[6,6,7,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, s, t], s, t, [[[s], [[[s, a0], 6], [[s, a1], 7]], 13], [[a0, s], [[[a0, t], 6], [[s, a1], 7]], 13], [[a1, s], [[[a1, t], 7], [[s, a0], 6]], 13], [[a0, a1, s], [[[a0, t], 6], [[a1, t], 7]], 13]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, s, t], s, t, [[[s], [[[s, a0], 6], [[s, a1], 7]], 13], [[a0, s], [[[a0, t], 6], [[s, a1], 7]], 13], [[a1, s], [[[a1, t], 7], [[s, a0], 6]], 13], [[a0, a1, s], [[[a0, t], 6], [[a1, t], 7]], 13]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155005,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[6,6,7,7],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[6,1],[7,1],[6,1],[7,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0006","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[7,7,8,8,9,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a0, s], [[[a0, t], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a1, s], [[[a1, t], 8], [[s, a0], 7], [[s, a2], 9]], 24], [[a0, a1, s], [[[a0, t], 7], [[a1, t], 8], [[s, a2], 9]], 24], [[a2, s], [[[a2, t], 9], [[s, a0], 7], [[s, a1], 8]], 24], [[a0, a2, s], [[[a0, t], 7], [[a2, t], 9], [[s, a1], 8]], 24], [[a1, a2, s], [[[a1, t], 8], [[a2, t], 9], [[s, a0], 7]], 24], [[a0, a1, a2, s], [[[a0, t], 7], [[a1, t], 8], [[a2, t], 9]], 24]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a0, s], [[[a0, t], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a1, s], [[[a1, t], 8], [[s, a0], 7], [[s, a2], 9]], 24], [[a0, a1, s], [[[a0, t], 7], [[a1, t], 8], [[s, a2], 9]], 24], [[a2, s], [[[a2, t], 9], [[s, a0], 7], [[s, a1], 8]], 24], [[a0, a2, s], [[[a0, t], 7], [[a2, t], 9], [[s, a1], 8]], 24], [[a1, a2, s], [[[a1, t], 8], [[a2, t], 9], [[s, a0], 7]], 24], [[a0, a1, a2, s], [[[a0, t], 7], [[a1, t], 8], [[a2, t], 9]], 24]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155006,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[7,7,8,8,9,9],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[7,1],[8,1],[9,1],[7,1],[8,1],[9,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0007","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[8,8,9,9,10,10,11,11,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, s], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a1, s], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, a1, s], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a2, s], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38], [[a0, a2, s], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38], [[a1, a2, s], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38], [[a0, a1, a2, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38], [[a3, s], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38], [[a0, a3, s], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38], [[a1, a3, s], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38], [[a0, a1, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38], [[a2, a3, s], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38], [[a0, a2, a3, s], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38], [[a1, a2, a3, s], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38], [[a0, a1, a2, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, s], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a1, s], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, a1, s], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a2, s], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38], [[a0, a2, s], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38], [[a1, a2, s], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38], [[a0, a1, a2, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38], [[a3, s], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38], [[a0, a3, s], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38], [[a1, a3, s], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38], [[a0, a1, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38], [[a2, a3, s], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38], [[a0, a2, a3, s], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38], [[a1, a2, a3, s], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38], [[a0, a1, a2, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155007,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[8,8,9,9,10,10,11,11,13],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[13,1],[8,1],[9,1],[10,1],[11,1],[8,1],[9,1],[10,1],[11,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0008","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[9,9,10,10,11,11,12,12,13,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155008,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[9,9,10,10,11,11,12,12,13,13],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[9,1],[10,1],[11,1],[12,1],[13,1],[9,1],[10,1],[11,1],[12,1],[13,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0009","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[10,10,11,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, s, t], s, t, [[[s], [[[s, a0], 10], [[s, a1], 11]], 21], [[a0, s], [[[a0, t], 10], [[s, a1], 11]], 21], [[a1, s], [[[a1, t], 11], [[s, a0], 10]], 21], [[a0, a1, s], [[[a0, t], 10], [[a1, t], 11]], 21]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, s, t], s, t, [[[s], [[[s, a0], 10], [[s, a1], 11]], 21], [[a0, s], [[[a0, t], 10], [[s, a1], 11]], 21], [[a1, s], [[[a1, t], 11], [[s, a0], 10]], 21], [[a0, a1, s], [[[a0, t], 10], [[a1, t], 11]], 21]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155009,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[10,10,11,11],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[10,1],[11,1],[10,1],[11,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0010","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[11,11,12,12,13,13,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a0, s], [[[a0, t], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a1, s], [[[a1, t], 12], [[s, a0], 11], [[s, a2], 13]], 36], [[a0, a1, s], [[[a0, t], 11], [[a1, t], 12], [[s, a2], 13]], 36], [[a2, s], [[[a2, t], 13], [[s, a0], 11], [[s, a1], 12]], 36], [[a0, a2, s], [[[a0, t], 11], [[a2, t], 13], [[s, a1], 12]], 36], [[a1, a2, s], [[[a1, t], 12], [[a2, t], 13], [[s, a0], 11]], 36], [[a0, a1, a2, s], [[[a0, t], 11], [[a1, t], 12], [[a2, t], 13]], 36]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a0, s], [[[a0, t], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a1, s], [[[a1, t], 12], [[s, a0], 11], [[s, a2], 13]], 36], [[a0, a1, s], [[[a0, t], 11], [[a1, t], 12], [[s, a2], 13]], 36], [[a2, s], [[[a2, t], 13], [[s, a0], 11], [[s, a1], 12]], 36], [[a0, a2, s], [[[a0, t], 11], [[a2, t], 13], [[s, a1], 12]], 36], [[a1, a2, s], [[[a1, t], 12], [[a2, t], 13], [[s, a0], 11]], 36], [[a0, a1, a2, s], [[[a0, t], 11], [[a1, t], 12], [[a2, t], 13]], 36]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155010,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[11,11,12,12,13,13,16],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[16,1],[11,1],[12,1],[13,1],[11,1],[12,1],[13,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0011","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[12,12,13,13,14,14,15,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, s], [[[a0, t], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a1, s], [[[a1, t], 13], [[s, a0], 12], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, a1, s], [[[a0, t], 12], [[a1, t], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a2, s], [[[a2, t], 14], [[s, a0], 12], [[s, a1], 13], [[s, a3], 15]], 54], [[a0, a2, s], [[[a0, t], 12], [[a2, t], 14], [[s, a1], 13], [[s, a3], 15]], 54], [[a1, a2, s], [[[a1, t], 13], [[a2, t], 14], [[s, a0], 12], [[s, a3], 15]], 54], [[a0, a1, a2, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[s, a3], 15]], 54], [[a3, s], [[[a3, t], 15], [[s, a0], 12], [[s, a1], 13], [[s, a2], 14]], 54], [[a0, a3, s], [[[a0, t], 12], [[a3, t], 15], [[s, a1], 13], [[s, a2], 14]], 54], [[a1, a3, s], [[[a1, t], 13], [[a3, t], 15], [[s, a0], 12], [[s, a2], 14]], 54], [[a0, a1, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a3, t], 15], [[s, a2], 14]], 54], [[a2, a3, s], [[[a2, t], 14], [[a3, t], 15], [[s, a0], 12], [[s, a1], 13]], 54], [[a0, a2, a3, s], [[[a0, t], 12], [[a2, t], 14], [[a3, t], 15], [[s, a1], 13]], 54], [[a1, a2, a3, s], [[[a1, t], 13], [[a2, t], 14], [[a3, t], 15], [[s, a0], 12]], 54], [[a0, a1, a2, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[a3, t], 15]], 54]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, s], [[[a0, t], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a1, s], [[[a1, t], 13], [[s, a0], 12], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, a1, s], [[[a0, t], 12], [[a1, t], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a2, s], [[[a2, t], 14], [[s, a0], 12], [[s, a1], 13], [[s, a3], 15]], 54], [[a0, a2, s], [[[a0, t], 12], [[a2, t], 14], [[s, a1], 13], [[s, a3], 15]], 54], [[a1, a2, s], [[[a1, t], 13], [[a2, t], 14], [[s, a0], 12], [[s, a3], 15]], 54], [[a0, a1, a2, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[s, a3], 15]], 54], [[a3, s], [[[a3, t], 15], [[s, a0], 12], [[s, a1], 13], [[s, a2], 14]], 54], [[a0, a3, s], [[[a0, t], 12], [[a3, t], 15], [[s, a1], 13], [[s, a2], 14]], 54], [[a1, a3, s], [[[a1, t], 13], [[a3, t], 15], [[s, a0], 12], [[s, a2], 14]], 54], [[a0, a1, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a3, t], 15], [[s, a2], 14]], 54], [[a2, a3, s], [[[a2, t], 14], [[a3, t], 15], [[s, a0], 12], [[s, a1], 13]], 54], [[a0, a2, a3, s], [[[a0, t], 12], [[a2, t], 14], [[a3, t], 15], [[s, a1], 13]], 54], [[a1, a2, a3, s], [[[a1, t], 13], [[a2, t], 14], [[a3, t], 15], [[s, a0], 12]], 54], [[a0, a1, a2, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[a3, t], 15]], 54]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155011,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[12,12,13,13,14,14,15,15],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[12,1],[13,1],[14,1],[15,1],[12,1],[13,1],[14,1],[15,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0012","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[13,13,14,14,15,15,16,16,17,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, s], [[[a0, t], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, s], [[[a1, t], 14], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, s], [[[a0, t], 13], [[a1, t], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a2, s], [[[a2, t], 15], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a2, s], [[[a0, t], 13], [[a2, t], 15], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, a2, s], [[[a1, t], 14], [[a2, t], 15], [[s, a0], 13], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, a2, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a3, s], [[[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a3, s], [[[a0, t], 13], [[a3, t], 16], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a1, a3, s], [[[a1, t], 14], [[a3, t], 16], [[s, a0], 13], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a1, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[s, a2], 15], [[s, a4], 17]], 75], [[a2, a3, s], [[[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a4], 17]], 75], [[a0, a2, a3, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[s, a1], 14], [[s, a4], 17]], 75], [[a1, a2, a3, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a4], 17]], 75], [[a0, a1, a2, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a4], 17]], 75], [[a4, s], [[[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a4, s], [[[a0, t], 13], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a1, a4, s], [[[a1, t], 14], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a1, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a4, t], 17], [[s, a2], 15], [[s, a3], 16]], 75], [[a2, a4, s], [[[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16]], 75], [[a0, a2, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a4, t], 17], [[s, a1], 14], [[s, a3], 16]], 75], [[a1, a2, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a3], 16]], 75], [[a0, a1, a2, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a3], 16]], 75], [[a3, a4, s], [[[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15]], 75], [[a0, a3, a4, s], [[[a0, t], 13], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15]], 75], [[a1, a3, a4, s], [[[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15]], 75], [[a0, a1, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a2], 15]], 75], [[a2, a3, a4, s], [[[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14]], 75], [[a0, a2, a3, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14]], 75], [[a1, a2, a3, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13]], 75], [[a0, a1, a2, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17]], 75]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, s], [[[a0, t], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, s], [[[a1, t], 14], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, s], [[[a0, t], 13], [[a1, t], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a2, s], [[[a2, t], 15], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a2, s], [[[a0, t], 13], [[a2, t], 15], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, a2, s], [[[a1, t], 14], [[a2, t], 15], [[s, a0], 13], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, a2, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a3, s], [[[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a3, s], [[[a0, t], 13], [[a3, t], 16], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a1, a3, s], [[[a1, t], 14], [[a3, t], 16], [[s, a0], 13], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a1, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[s, a2], 15], [[s, a4], 17]], 75], [[a2, a3, s], [[[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a4], 17]], 75], [[a0, a2, a3, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[s, a1], 14], [[s, a4], 17]], 75], [[a1, a2, a3, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a4], 17]], 75], [[a0, a1, a2, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a4], 17]], 75], [[a4, s], [[[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a4, s], [[[a0, t], 13], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a1, a4, s], [[[a1, t], 14], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a1, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a4, t], 17], [[s, a2], 15], [[s, a3], 16]], 75], [[a2, a4, s], [[[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16]], 75], [[a0, a2, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a4, t], 17], [[s, a1], 14], [[s, a3], 16]], 75], [[a1, a2, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a3], 16]], 75], [[a0, a1, a2, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a3], 16]], 75], [[a3, a4, s], [[[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15]], 75], [[a0, a3, a4, s], [[[a0, t], 13], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15]], 75], [[a1, a3, a4, s], [[[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15]], 75], [[a0, a1, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a2], 15]], 75], [[a2, a3, a4, s], [[[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14]], 75], [[a0, a2, a3, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14]], 75], [[a1, a2, a3, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13]], 75], [[a0, a1, a2, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17]], 75]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155012,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[13,13,14,14,15,15,16,16,17,17],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[13,1],[14,1],[15,1],[16,1],[17,1],[13,1],[14,1],[15,1],[16,1],[17,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0013","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[14,14,15,15,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, s, t], s, t, [[[s], [[[s, a0], 14], [[s, a1], 15]], 29], [[a0, s], [[[a0, t], 14], [[s, a1], 15]], 29], [[a1, s], [[[a1, t], 15], [[s, a0], 14]], 29], [[a0, a1, s], [[[a0, t], 14], [[a1, t], 15]], 29]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, s, t], s, t, [[[s], [[[s, a0], 14], [[s, a1], 15]], 29], [[a0, s], [[[a0, t], 14], [[s, a1], 15]], 29], [[a1, s], [[[a1, t], 15], [[s, a0], 14]], 29], [[a0, a1, s], [[[a0, t], 14], [[a1, t], 15]], 29]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155013,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[14,14,15,15,19],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[19,1],[14,1],[15,1],[14,1],[15,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0014","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[15,15,16,16,17,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a0, s], [[[a0, t], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a1, s], [[[a1, t], 16], [[s, a0], 15], [[s, a2], 17]], 48], [[a0, a1, s], [[[a0, t], 15], [[a1, t], 16], [[s, a2], 17]], 48], [[a2, s], [[[a2, t], 17], [[s, a0], 15], [[s, a1], 16]], 48], [[a0, a2, s], [[[a0, t], 15], [[a2, t], 17], [[s, a1], 16]], 48], [[a1, a2, s], [[[a1, t], 16], [[a2, t], 17], [[s, a0], 15]], 48], [[a0, a1, a2, s], [[[a0, t], 15], [[a1, t], 16], [[a2, t], 17]], 48]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a0, s], [[[a0, t], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a1, s], [[[a1, t], 16], [[s, a0], 15], [[s, a2], 17]], 48], [[a0, a1, s], [[[a0, t], 15], [[a1, t], 16], [[s, a2], 17]], 48], [[a2, s], [[[a2, t], 17], [[s, a0], 15], [[s, a1], 16]], 48], [[a0, a2, s], [[[a0, t], 15], [[a2, t], 17], [[s, a1], 16]], 48], [[a1, a2, s], [[[a1, t], 16], [[a2, t], 17], [[s, a0], 15]], 48], [[a0, a1, a2, s], [[[a0, t], 15], [[a1, t], 16], [[a2, t], 17]], 48]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155014,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[15,15,16,16,17,17],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[15,1],[16,1],[17,1],[15,1],[16,1],[17,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0015","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[16,16,17,17,18,18,19,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, s], [[[a0, t], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a1, s], [[[a1, t], 17], [[s, a0], 16], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, a1, s], [[[a0, t], 16], [[a1, t], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a2, s], [[[a2, t], 18], [[s, a0], 16], [[s, a1], 17], [[s, a3], 19]], 70], [[a0, a2, s], [[[a0, t], 16], [[a2, t], 18], [[s, a1], 17], [[s, a3], 19]], 70], [[a1, a2, s], [[[a1, t], 17], [[a2, t], 18], [[s, a0], 16], [[s, a3], 19]], 70], [[a0, a1, a2, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[s, a3], 19]], 70], [[a3, s], [[[a3, t], 19], [[s, a0], 16], [[s, a1], 17], [[s, a2], 18]], 70], [[a0, a3, s], [[[a0, t], 16], [[a3, t], 19], [[s, a1], 17], [[s, a2], 18]], 70], [[a1, a3, s], [[[a1, t], 17], [[a3, t], 19], [[s, a0], 16], [[s, a2], 18]], 70], [[a0, a1, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a3, t], 19], [[s, a2], 18]], 70], [[a2, a3, s], [[[a2, t], 18], [[a3, t], 19], [[s, a0], 16], [[s, a1], 17]], 70], [[a0, a2, a3, s], [[[a0, t], 16], [[a2, t], 18], [[a3, t], 19], [[s, a1], 17]], 70], [[a1, a2, a3, s], [[[a1, t], 17], [[a2, t], 18], [[a3, t], 19], [[s, a0], 16]], 70], [[a0, a1, a2, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[a3, t], 19]], 70]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, s], [[[a0, t], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a1, s], [[[a1, t], 17], [[s, a0], 16], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, a1, s], [[[a0, t], 16], [[a1, t], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a2, s], [[[a2, t], 18], [[s, a0], 16], [[s, a1], 17], [[s, a3], 19]], 70], [[a0, a2, s], [[[a0, t], 16], [[a2, t], 18], [[s, a1], 17], [[s, a3], 19]], 70], [[a1, a2, s], [[[a1, t], 17], [[a2, t], 18], [[s, a0], 16], [[s, a3], 19]], 70], [[a0, a1, a2, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[s, a3], 19]], 70], [[a3, s], [[[a3, t], 19], [[s, a0], 16], [[s, a1], 17], [[s, a2], 18]], 70], [[a0, a3, s], [[[a0, t], 16], [[a3, t], 19], [[s, a1], 17], [[s, a2], 18]], 70], [[a1, a3, s], [[[a1, t], 17], [[a3, t], 19], [[s, a0], 16], [[s, a2], 18]], 70], [[a0, a1, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a3, t], 19], [[s, a2], 18]], 70], [[a2, a3, s], [[[a2, t], 18], [[a3, t], 19], [[s, a0], 16], [[s, a1], 17]], 70], [[a0, a2, a3, s], [[[a0, t], 16], [[a2, t], 18], [[a3, t], 19], [[s, a1], 17]], 70], [[a1, a2, a3, s], [[[a1, t], 17], [[a2, t], 18], [[a3, t], 19], [[s, a0], 16]], 70], [[a0, a1, a2, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[a3, t], 19]], 70]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155015,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[16,16,17,17,18,18,19,19],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[16,1],[17,1],[18,1],[19,1],[16,1],[17,1],[18,1],[19,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0016","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[17,17,18,18,19,19,20,20,21,21,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, s], [[[a0, t], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, s], [[[a1, t], 18], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, s], [[[a0, t], 17], [[a1, t], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a2, s], [[[a2, t], 19], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a2, s], [[[a0, t], 17], [[a2, t], 19], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, a2, s], [[[a1, t], 18], [[a2, t], 19], [[s, a0], 17], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, a2, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a3, s], [[[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a3, s], [[[a0, t], 17], [[a3, t], 20], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a1, a3, s], [[[a1, t], 18], [[a3, t], 20], [[s, a0], 17], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a1, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[s, a2], 19], [[s, a4], 21]], 95], [[a2, a3, s], [[[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a4], 21]], 95], [[a0, a2, a3, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[s, a1], 18], [[s, a4], 21]], 95], [[a1, a2, a3, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a4], 21]], 95], [[a0, a1, a2, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a4], 21]], 95], [[a4, s], [[[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a4, s], [[[a0, t], 17], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a1, a4, s], [[[a1, t], 18], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a1, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a4, t], 21], [[s, a2], 19], [[s, a3], 20]], 95], [[a2, a4, s], [[[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20]], 95], [[a0, a2, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a4, t], 21], [[s, a1], 18], [[s, a3], 20]], 95], [[a1, a2, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a3], 20]], 95], [[a0, a1, a2, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a3], 20]], 95], [[a3, a4, s], [[[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19]], 95], [[a0, a3, a4, s], [[[a0, t], 17], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19]], 95], [[a1, a3, a4, s], [[[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19]], 95], [[a0, a1, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a2], 19]], 95], [[a2, a3, a4, s], [[[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18]], 95], [[a0, a2, a3, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18]], 95], [[a1, a2, a3, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17]], 95], [[a0, a1, a2, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21]], 95]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, s], [[[a0, t], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, s], [[[a1, t], 18], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, s], [[[a0, t], 17], [[a1, t], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a2, s], [[[a2, t], 19], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a2, s], [[[a0, t], 17], [[a2, t], 19], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, a2, s], [[[a1, t], 18], [[a2, t], 19], [[s, a0], 17], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, a2, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a3, s], [[[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a3, s], [[[a0, t], 17], [[a3, t], 20], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a1, a3, s], [[[a1, t], 18], [[a3, t], 20], [[s, a0], 17], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a1, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[s, a2], 19], [[s, a4], 21]], 95], [[a2, a3, s], [[[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a4], 21]], 95], [[a0, a2, a3, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[s, a1], 18], [[s, a4], 21]], 95], [[a1, a2, a3, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a4], 21]], 95], [[a0, a1, a2, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a4], 21]], 95], [[a4, s], [[[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a4, s], [[[a0, t], 17], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a1, a4, s], [[[a1, t], 18], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a1, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a4, t], 21], [[s, a2], 19], [[s, a3], 20]], 95], [[a2, a4, s], [[[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20]], 95], [[a0, a2, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a4, t], 21], [[s, a1], 18], [[s, a3], 20]], 95], [[a1, a2, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a3], 20]], 95], [[a0, a1, a2, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a3], 20]], 95], [[a3, a4, s], [[[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19]], 95], [[a0, a3, a4, s], [[[a0, t], 17], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19]], 95], [[a1, a3, a4, s], [[[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19]], 95], [[a0, a1, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a2], 19]], 95], [[a2, a3, a4, s], [[[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18]], 95], [[a0, a2, a3, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18]], 95], [[a1, a2, a3, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17]], 95], [[a0, a1, a2, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21]], 95]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155016,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"],["a0","s"]],"source":"s","sink":"t","capacities":[17,17,18,18,19,19,20,20,21,21,22],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[22,1],[17,1],[18,1],[19,1],[20,1],[21,1],[17,1],[18,1],[19,1],[20,1],[21,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0017","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[18,18,19,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, s, t], s, t, [[[s], [[[s, a0], 18], [[s, a1], 19]], 37], [[a0, s], [[[a0, t], 18], [[s, a1], 19]], 37], [[a1, s], [[[a1, t], 19], [[s, a0], 18]], 37], [[a0, a1, s], [[[a0, t], 18], [[a1, t], 19]], 37]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, s, t], s, t, [[[s], [[[s, a0], 18], [[s, a1], 19]], 37], [[a0, s], [[[a0, t], 18], [[s, a1], 19]], 37], [[a1, s], [[[a1, t], 19], [[s, a0], 18]], 37], [[a0, a1, s], [[[a0, t], 18], [[a1, t], 19]], 37]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155017,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[18,18,19,19],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[18,1],[19,1],[18,1],[19,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0018","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[19,19,20,20,21,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a0, s], [[[a0, t], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a1, s], [[[a1, t], 20], [[s, a0], 19], [[s, a2], 21]], 60], [[a0, a1, s], [[[a0, t], 19], [[a1, t], 20], [[s, a2], 21]], 60], [[a2, s], [[[a2, t], 21], [[s, a0], 19], [[s, a1], 20]], 60], [[a0, a2, s], [[[a0, t], 19], [[a2, t], 21], [[s, a1], 20]], 60], [[a1, a2, s], [[[a1, t], 20], [[a2, t], 21], [[s, a0], 19]], 60], [[a0, a1, a2, s], [[[a0, t], 19], [[a1, t], 20], [[a2, t], 21]], 60]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a0, s], [[[a0, t], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a1, s], [[[a1, t], 20], [[s, a0], 19], [[s, a2], 21]], 60], [[a0, a1, s], [[[a0, t], 19], [[a1, t], 20], [[s, a2], 21]], 60], [[a2, s], [[[a2, t], 21], [[s, a0], 19], [[s, a1], 20]], 60], [[a0, a2, s], [[[a0, t], 19], [[a2, t], 21], [[s, a1], 20]], 60], [[a1, a2, s], [[[a1, t], 20], [[a2, t], 21], [[s, a0], 19]], 60], [[a0, a1, a2, s], [[[a0, t], 19], [[a1, t], 20], [[a2, t], 21]], 60]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155018,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[19,19,20,20,21,21],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[19,1],[20,1],[21,1],[19,1],[20,1],[21,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0019","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[20,20,21,21,22,22,23,23,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, s], [[[a0, t], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a1, s], [[[a1, t], 21], [[s, a0], 20], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, a1, s], [[[a0, t], 20], [[a1, t], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a2, s], [[[a2, t], 22], [[s, a0], 20], [[s, a1], 21], [[s, a3], 23]], 86], [[a0, a2, s], [[[a0, t], 20], [[a2, t], 22], [[s, a1], 21], [[s, a3], 23]], 86], [[a1, a2, s], [[[a1, t], 21], [[a2, t], 22], [[s, a0], 20], [[s, a3], 23]], 86], [[a0, a1, a2, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[s, a3], 23]], 86], [[a3, s], [[[a3, t], 23], [[s, a0], 20], [[s, a1], 21], [[s, a2], 22]], 86], [[a0, a3, s], [[[a0, t], 20], [[a3, t], 23], [[s, a1], 21], [[s, a2], 22]], 86], [[a1, a3, s], [[[a1, t], 21], [[a3, t], 23], [[s, a0], 20], [[s, a2], 22]], 86], [[a0, a1, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a3, t], 23], [[s, a2], 22]], 86], [[a2, a3, s], [[[a2, t], 22], [[a3, t], 23], [[s, a0], 20], [[s, a1], 21]], 86], [[a0, a2, a3, s], [[[a0, t], 20], [[a2, t], 22], [[a3, t], 23], [[s, a1], 21]], 86], [[a1, a2, a3, s], [[[a1, t], 21], [[a2, t], 22], [[a3, t], 23], [[s, a0], 20]], 86], [[a0, a1, a2, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[a3, t], 23]], 86]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, s], [[[a0, t], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a1, s], [[[a1, t], 21], [[s, a0], 20], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, a1, s], [[[a0, t], 20], [[a1, t], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a2, s], [[[a2, t], 22], [[s, a0], 20], [[s, a1], 21], [[s, a3], 23]], 86], [[a0, a2, s], [[[a0, t], 20], [[a2, t], 22], [[s, a1], 21], [[s, a3], 23]], 86], [[a1, a2, s], [[[a1, t], 21], [[a2, t], 22], [[s, a0], 20], [[s, a3], 23]], 86], [[a0, a1, a2, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[s, a3], 23]], 86], [[a3, s], [[[a3, t], 23], [[s, a0], 20], [[s, a1], 21], [[s, a2], 22]], 86], [[a0, a3, s], [[[a0, t], 20], [[a3, t], 23], [[s, a1], 21], [[s, a2], 22]], 86], [[a1, a3, s], [[[a1, t], 21], [[a3, t], 23], [[s, a0], 20], [[s, a2], 22]], 86], [[a0, a1, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a3, t], 23], [[s, a2], 22]], 86], [[a2, a3, s], [[[a2, t], 22], [[a3, t], 23], [[s, a0], 20], [[s, a1], 21]], 86], [[a0, a2, a3, s], [[[a0, t], 20], [[a2, t], 22], [[a3, t], 23], [[s, a1], 21]], 86], [[a1, a2, a3, s], [[[a1, t], 21], [[a2, t], 22], [[a3, t], 23], [[s, a0], 20]], 86], [[a0, a1, a2, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[a3, t], 23]], 86]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155019,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[20,20,21,21,22,22,23,23,25],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[25,1],[20,1],[21,1],[22,1],[23,1],[20,1],[21,1],[22,1],[23,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0020","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[21,21,22,22,23,23,24,24,25,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, s], [[[a0, t], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, s], [[[a1, t], 22], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, s], [[[a0, t], 21], [[a1, t], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a2, s], [[[a2, t], 23], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a2, s], [[[a0, t], 21], [[a2, t], 23], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, a2, s], [[[a1, t], 22], [[a2, t], 23], [[s, a0], 21], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, a2, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a3, s], [[[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a3, s], [[[a0, t], 21], [[a3, t], 24], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a1, a3, s], [[[a1, t], 22], [[a3, t], 24], [[s, a0], 21], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a1, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[s, a2], 23], [[s, a4], 25]], 115], [[a2, a3, s], [[[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a4], 25]], 115], [[a0, a2, a3, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[s, a1], 22], [[s, a4], 25]], 115], [[a1, a2, a3, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a4], 25]], 115], [[a0, a1, a2, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a4], 25]], 115], [[a4, s], [[[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a4, s], [[[a0, t], 21], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a1, a4, s], [[[a1, t], 22], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a1, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a4, t], 25], [[s, a2], 23], [[s, a3], 24]], 115], [[a2, a4, s], [[[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24]], 115], [[a0, a2, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a4, t], 25], [[s, a1], 22], [[s, a3], 24]], 115], [[a1, a2, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a3], 24]], 115], [[a0, a1, a2, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a3], 24]], 115], [[a3, a4, s], [[[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23]], 115], [[a0, a3, a4, s], [[[a0, t], 21], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23]], 115], [[a1, a3, a4, s], [[[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23]], 115], [[a0, a1, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a2], 23]], 115], [[a2, a3, a4, s], [[[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22]], 115], [[a0, a2, a3, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22]], 115], [[a1, a2, a3, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21]], 115], [[a0, a1, a2, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25]], 115]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, s], [[[a0, t], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, s], [[[a1, t], 22], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, s], [[[a0, t], 21], [[a1, t], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a2, s], [[[a2, t], 23], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a2, s], [[[a0, t], 21], [[a2, t], 23], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, a2, s], [[[a1, t], 22], [[a2, t], 23], [[s, a0], 21], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, a2, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a3, s], [[[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a3, s], [[[a0, t], 21], [[a3, t], 24], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a1, a3, s], [[[a1, t], 22], [[a3, t], 24], [[s, a0], 21], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a1, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[s, a2], 23], [[s, a4], 25]], 115], [[a2, a3, s], [[[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a4], 25]], 115], [[a0, a2, a3, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[s, a1], 22], [[s, a4], 25]], 115], [[a1, a2, a3, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a4], 25]], 115], [[a0, a1, a2, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a4], 25]], 115], [[a4, s], [[[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a4, s], [[[a0, t], 21], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a1, a4, s], [[[a1, t], 22], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a1, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a4, t], 25], [[s, a2], 23], [[s, a3], 24]], 115], [[a2, a4, s], [[[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24]], 115], [[a0, a2, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a4, t], 25], [[s, a1], 22], [[s, a3], 24]], 115], [[a1, a2, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a3], 24]], 115], [[a0, a1, a2, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a3], 24]], 115], [[a3, a4, s], [[[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23]], 115], [[a0, a3, a4, s], [[[a0, t], 21], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23]], 115], [[a1, a3, a4, s], [[[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23]], 115], [[a0, a1, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a2], 23]], 115], [[a2, a3, a4, s], [[[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22]], 115], [[a0, a2, a3, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22]], 115], [[a1, a2, a3, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21]], 115], [[a0, a1, a2, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25]], 115]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155020,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[21,21,22,22,23,23,24,24,25,25],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[21,1],[22,1],[23,1],[24,1],[25,1],[21,1],[22,1],[23,1],[24,1],[25,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0021","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[22,22,23,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, s, t], s, t, [[[s], [[[s, a0], 22], [[s, a1], 23]], 45], [[a0, s], [[[a0, t], 22], [[s, a1], 23]], 45], [[a1, s], [[[a1, t], 23], [[s, a0], 22]], 45], [[a0, a1, s], [[[a0, t], 22], [[a1, t], 23]], 45]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, s, t], s, t, [[[s], [[[s, a0], 22], [[s, a1], 23]], 45], [[a0, s], [[[a0, t], 22], [[s, a1], 23]], 45], [[a1, s], [[[a1, t], 23], [[s, a0], 22]], 45], [[a0, a1, s], [[[a0, t], 22], [[a1, t], 23]], 45]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155021,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[22,22,23,23],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[22,1],[23,1],[22,1],[23,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0022","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[23,23,24,24,25,25,28]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a0, s], [[[a0, t], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a1, s], [[[a1, t], 24], [[s, a0], 23], [[s, a2], 25]], 72], [[a0, a1, s], [[[a0, t], 23], [[a1, t], 24], [[s, a2], 25]], 72], [[a2, s], [[[a2, t], 25], [[s, a0], 23], [[s, a1], 24]], 72], [[a0, a2, s], [[[a0, t], 23], [[a2, t], 25], [[s, a1], 24]], 72], [[a1, a2, s], [[[a1, t], 24], [[a2, t], 25], [[s, a0], 23]], 72], [[a0, a1, a2, s], [[[a0, t], 23], [[a1, t], 24], [[a2, t], 25]], 72]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a0, s], [[[a0, t], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a1, s], [[[a1, t], 24], [[s, a0], 23], [[s, a2], 25]], 72], [[a0, a1, s], [[[a0, t], 23], [[a1, t], 24], [[s, a2], 25]], 72], [[a2, s], [[[a2, t], 25], [[s, a0], 23], [[s, a1], 24]], 72], [[a0, a2, s], [[[a0, t], 23], [[a2, t], 25], [[s, a1], 24]], 72], [[a1, a2, s], [[[a1, t], 24], [[a2, t], 25], [[s, a0], 23]], 72], [[a0, a1, a2, s], [[[a0, t], 23], [[a1, t], 24], [[a2, t], 25]], 72]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155022,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[23,23,24,24,25,25,28],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[28,1],[23,1],[24,1],[25,1],[23,1],[24,1],[25,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0023","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[24,24,25,25,26,26,27,27]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, s], [[[a0, t], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a1, s], [[[a1, t], 25], [[s, a0], 24], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, a1, s], [[[a0, t], 24], [[a1, t], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a2, s], [[[a2, t], 26], [[s, a0], 24], [[s, a1], 25], [[s, a3], 27]], 102], [[a0, a2, s], [[[a0, t], 24], [[a2, t], 26], [[s, a1], 25], [[s, a3], 27]], 102], [[a1, a2, s], [[[a1, t], 25], [[a2, t], 26], [[s, a0], 24], [[s, a3], 27]], 102], [[a0, a1, a2, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[s, a3], 27]], 102], [[a3, s], [[[a3, t], 27], [[s, a0], 24], [[s, a1], 25], [[s, a2], 26]], 102], [[a0, a3, s], [[[a0, t], 24], [[a3, t], 27], [[s, a1], 25], [[s, a2], 26]], 102], [[a1, a3, s], [[[a1, t], 25], [[a3, t], 27], [[s, a0], 24], [[s, a2], 26]], 102], [[a0, a1, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a3, t], 27], [[s, a2], 26]], 102], [[a2, a3, s], [[[a2, t], 26], [[a3, t], 27], [[s, a0], 24], [[s, a1], 25]], 102], [[a0, a2, a3, s], [[[a0, t], 24], [[a2, t], 26], [[a3, t], 27], [[s, a1], 25]], 102], [[a1, a2, a3, s], [[[a1, t], 25], [[a2, t], 26], [[a3, t], 27], [[s, a0], 24]], 102], [[a0, a1, a2, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[a3, t], 27]], 102]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, s], [[[a0, t], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a1, s], [[[a1, t], 25], [[s, a0], 24], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, a1, s], [[[a0, t], 24], [[a1, t], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a2, s], [[[a2, t], 26], [[s, a0], 24], [[s, a1], 25], [[s, a3], 27]], 102], [[a0, a2, s], [[[a0, t], 24], [[a2, t], 26], [[s, a1], 25], [[s, a3], 27]], 102], [[a1, a2, s], [[[a1, t], 25], [[a2, t], 26], [[s, a0], 24], [[s, a3], 27]], 102], [[a0, a1, a2, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[s, a3], 27]], 102], [[a3, s], [[[a3, t], 27], [[s, a0], 24], [[s, a1], 25], [[s, a2], 26]], 102], [[a0, a3, s], [[[a0, t], 24], [[a3, t], 27], [[s, a1], 25], [[s, a2], 26]], 102], [[a1, a3, s], [[[a1, t], 25], [[a3, t], 27], [[s, a0], 24], [[s, a2], 26]], 102], [[a0, a1, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a3, t], 27], [[s, a2], 26]], 102], [[a2, a3, s], [[[a2, t], 26], [[a3, t], 27], [[s, a0], 24], [[s, a1], 25]], 102], [[a0, a2, a3, s], [[[a0, t], 24], [[a2, t], 26], [[a3, t], 27], [[s, a1], 25]], 102], [[a1, a2, a3, s], [[[a1, t], 25], [[a2, t], 26], [[a3, t], 27], [[s, a0], 24]], 102], [[a0, a1, a2, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[a3, t], 27]], 102]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155023,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[24,24,25,25,26,26,27,27],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[24,1],[25,1],[26,1],[27,1],[24,1],[25,1],[26,1],[27,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0024","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[25,25,26,26,27,27,28,28,29,29]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, s], [[[a0, t], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, s], [[[a1, t], 26], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, s], [[[a0, t], 25], [[a1, t], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a2, s], [[[a2, t], 27], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a2, s], [[[a0, t], 25], [[a2, t], 27], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, a2, s], [[[a1, t], 26], [[a2, t], 27], [[s, a0], 25], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, a2, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a3, s], [[[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a3, s], [[[a0, t], 25], [[a3, t], 28], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a1, a3, s], [[[a1, t], 26], [[a3, t], 28], [[s, a0], 25], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a1, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[s, a2], 27], [[s, a4], 29]], 135], [[a2, a3, s], [[[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a4], 29]], 135], [[a0, a2, a3, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[s, a1], 26], [[s, a4], 29]], 135], [[a1, a2, a3, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a4], 29]], 135], [[a0, a1, a2, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a4], 29]], 135], [[a4, s], [[[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a4, s], [[[a0, t], 25], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a1, a4, s], [[[a1, t], 26], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a1, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a4, t], 29], [[s, a2], 27], [[s, a3], 28]], 135], [[a2, a4, s], [[[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28]], 135], [[a0, a2, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a4, t], 29], [[s, a1], 26], [[s, a3], 28]], 135], [[a1, a2, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a3], 28]], 135], [[a0, a1, a2, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a3], 28]], 135], [[a3, a4, s], [[[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27]], 135], [[a0, a3, a4, s], [[[a0, t], 25], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27]], 135], [[a1, a3, a4, s], [[[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27]], 135], [[a0, a1, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a2], 27]], 135], [[a2, a3, a4, s], [[[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26]], 135], [[a0, a2, a3, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26]], 135], [[a1, a2, a3, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25]], 135], [[a0, a1, a2, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29]], 135]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, s], [[[a0, t], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, s], [[[a1, t], 26], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, s], [[[a0, t], 25], [[a1, t], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a2, s], [[[a2, t], 27], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a2, s], [[[a0, t], 25], [[a2, t], 27], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, a2, s], [[[a1, t], 26], [[a2, t], 27], [[s, a0], 25], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, a2, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a3, s], [[[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a3, s], [[[a0, t], 25], [[a3, t], 28], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a1, a3, s], [[[a1, t], 26], [[a3, t], 28], [[s, a0], 25], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a1, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[s, a2], 27], [[s, a4], 29]], 135], [[a2, a3, s], [[[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a4], 29]], 135], [[a0, a2, a3, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[s, a1], 26], [[s, a4], 29]], 135], [[a1, a2, a3, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a4], 29]], 135], [[a0, a1, a2, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a4], 29]], 135], [[a4, s], [[[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a4, s], [[[a0, t], 25], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a1, a4, s], [[[a1, t], 26], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a1, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a4, t], 29], [[s, a2], 27], [[s, a3], 28]], 135], [[a2, a4, s], [[[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28]], 135], [[a0, a2, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a4, t], 29], [[s, a1], 26], [[s, a3], 28]], 135], [[a1, a2, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a3], 28]], 135], [[a0, a1, a2, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a3], 28]], 135], [[a3, a4, s], [[[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27]], 135], [[a0, a3, a4, s], [[[a0, t], 25], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27]], 135], [[a1, a3, a4, s], [[[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27]], 135], [[a0, a1, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a2], 27]], 135], [[a2, a3, a4, s], [[[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26]], 135], [[a0, a2, a3, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26]], 135], [[a1, a2, a3, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25]], 135], [[a0, a1, a2, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29]], 135]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155024,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[25,25,26,26,27,27,28,28,29,29],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[25,1],[26,1],[27,1],[28,1],[29,1],[25,1],[26,1],[27,1],[28,1],[29,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0025","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[26,26,27,27,31]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, a1, s, t], s, t, [[[s], [[[s, a0], 26], [[s, a1], 27]], 53], [[a0, s], [[[a0, t], 26], [[s, a1], 27]], 53], [[a1, s], [[[a1, t], 27], [[s, a0], 26]], 53], [[a0, a1, s], [[[a0, t], 26], [[a1, t], 27]], 53]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)","explanation_zh":"Wave6第九層核對=([a0, a1, s, t], s, t, [[[s], [[[s, a0], 26], [[s, a1], 27]], 53], [[a0, s], [[[a0, t], 26], [[s, a1], 27]], 53], [[a1, s], [[[a1, t], 27], [[s, a0], 26]], 53], [[a0, a1, s], [[[a0, t], 26], [[a1, t], 27]], 53]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155025,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[26,26,27,27,31],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-complete-cut-enumeration","semantic_template":"qf-mod46-mfmc-complete-cut-enumeration/{\"capacities\":[[31,1],[26,1],[27,1],[26,1],[27,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0026","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[2,2,3,3,7],\"flows\":[1,1,2,2,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, [[[s], 3, 0, 3, 5, 是, 是], [[a0, s], 3, 0, 3, 5, 是, 是], [[a1, s], 3, 0, 3, 5, 是, 是], [[a0, a1, s], 3, 0, 3, 5, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(3, 3, [[[s], 3, 0, 3, 5, 是, 是], [[a0, s], 3, 0, 3, 5, 是, 是], [[a1, s], 3, 0, 3, 5, 是, 是], [[a0, a1, s], 3, 0, 3, 5, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155026,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[2,2,3,3,7],"flows":[1,1,2,2,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[7,1],[2,1],[3,1],[2,1],[3,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[0,1],[1,1],[2,1],[1,1],[2,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0027","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[3,3,4,4,5,5],\"flows\":[2,2,3,3,4,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 9, [[[s], 9, 0, 9, 12, 是, 是], [[a0, s], 9, 0, 9, 12, 是, 是], [[a1, s], 9, 0, 9, 12, 是, 是], [[a0, a1, s], 9, 0, 9, 12, 是, 是], [[a2, s], 9, 0, 9, 12, 是, 是], [[a0, a2, s], 9, 0, 9, 12, 是, 是], [[a1, a2, s], 9, 0, 9, 12, 是, 是], [[a0, a1, a2, s], 9, 0, 9, 12, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(9, 9, [[[s], 9, 0, 9, 12, 是, 是], [[a0, s], 9, 0, 9, 12, 是, 是], [[a1, s], 9, 0, 9, 12, 是, 是], [[a0, a1, s], 9, 0, 9, 12, 是, 是], [[a2, s], 9, 0, 9, 12, 是, 是], [[a0, a2, s], 9, 0, 9, 12, 是, 是], [[a1, a2, s], 9, 0, 9, 12, 是, 是], [[a0, a1, a2, s], 9, 0, 9, 12, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155027,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[3,3,4,4,5,5],"flows":[2,2,3,3,4,4],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[3,1],[4,1],[5,1],[3,1],[4,1],[5,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"flows\":[[2,1],[3,1],[4,1],[2,1],[3,1],[4,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0028","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[4,4,5,5,6,6,7,7],\"flows\":[3,3,4,4,5,5,6,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 18, [[[s], 18, 0, 18, 22, 是, 是], [[a0, s], 18, 0, 18, 22, 是, 是], [[a1, s], 18, 0, 18, 22, 是, 是], [[a0, a1, s], 18, 0, 18, 22, 是, 是], [[a2, s], 18, 0, 18, 22, 是, 是], [[a0, a2, s], 18, 0, 18, 22, 是, 是], [[a1, a2, s], 18, 0, 18, 22, 是, 是], [[a0, a1, a2, s], 18, 0, 18, 22, 是, 是], [[a3, s], 18, 0, 18, 22, 是, 是], [[a0, a3, s], 18, 0, 18, 22, 是, 是], [[a1, a3, s], 18, 0, 18, 22, 是, 是], [[a0, a1, a3, s], 18, 0, 18, 22, 是, 是], [[a2, a3, s], 18, 0, 18, 22, 是, 是], [[a0, a2, a3, s], 18, 0, 18, 22, 是, 是], [[a1, a2, a3, s], 18, 0, 18, 22, 是, 是], [[a0, a1, a2, a3, s], 18, 0, 18, 22, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(18, 18, [[[s], 18, 0, 18, 22, 是, 是], [[a0, s], 18, 0, 18, 22, 是, 是], [[a1, s], 18, 0, 18, 22, 是, 是], [[a0, a1, s], 18, 0, 18, 22, 是, 是], [[a2, s], 18, 0, 18, 22, 是, 是], [[a0, a2, s], 18, 0, 18, 22, 是, 是], [[a1, a2, s], 18, 0, 18, 22, 是, 是], [[a0, a1, a2, s], 18, 0, 18, 22, 是, 是], [[a3, s], 18, 0, 18, 22, 是, 是], [[a0, a3, s], 18, 0, 18, 22, 是, 是], [[a1, a3, s], 18, 0, 18, 22, 是, 是], [[a0, a1, a3, s], 18, 0, 18, 22, 是, 是], [[a2, a3, s], 18, 0, 18, 22, 是, 是], [[a0, a2, a3, s], 18, 0, 18, 22, 是, 是], [[a1, a2, a3, s], 18, 0, 18, 22, 是, 是], [[a0, a1, a2, a3, s], 18, 0, 18, 22, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155028,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[4,4,5,5,6,6,7,7],"flows":[3,3,4,4,5,5,6,6],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[4,1],[5,1],[6,1],[7,1],[4,1],[5,1],[6,1],[7,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"flows\":[[3,1],[4,1],[5,1],[6,1],[3,1],[4,1],[5,1],[6,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0029","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[5,5,6,6,7,7,8,8,9,9,10],\"flows\":[4,4,5,5,6,6,7,7,8,8,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(30, 30, [[[s], 30, 0, 30, 35, 是, 是], [[a0, s], 30, 0, 30, 35, 是, 是], [[a1, s], 30, 0, 30, 35, 是, 是], [[a0, a1, s], 30, 0, 30, 35, 是, 是], [[a2, s], 30, 0, 30, 35, 是, 是], [[a0, a2, s], 30, 0, 30, 35, 是, 是], [[a1, a2, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, s], 30, 0, 30, 35, 是, 是], [[a3, s], 30, 0, 30, 35, 是, 是], [[a0, a3, s], 30, 0, 30, 35, 是, 是], [[a1, a3, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a3, s], 30, 0, 30, 35, 是, 是], [[a2, a3, s], 30, 0, 30, 35, 是, 是], [[a0, a2, a3, s], 30, 0, 30, 35, 是, 是], [[a1, a2, a3, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, a3, s], 30, 0, 30, 35, 是, 是], [[a4, s], 30, 0, 30, 35, 是, 是], [[a0, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a4, s], 30, 0, 30, 35, 是, 是], [[a2, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a2, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a2, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, a4, s], 30, 0, 30, 35, 是, 是], [[a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a2, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a2, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a2, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, a3, a4, s], 30, 0, 30, 35, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(30, 30, [[[s], 30, 0, 30, 35, 是, 是], [[a0, s], 30, 0, 30, 35, 是, 是], [[a1, s], 30, 0, 30, 35, 是, 是], [[a0, a1, s], 30, 0, 30, 35, 是, 是], [[a2, s], 30, 0, 30, 35, 是, 是], [[a0, a2, s], 30, 0, 30, 35, 是, 是], [[a1, a2, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, s], 30, 0, 30, 35, 是, 是], [[a3, s], 30, 0, 30, 35, 是, 是], [[a0, a3, s], 30, 0, 30, 35, 是, 是], [[a1, a3, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a3, s], 30, 0, 30, 35, 是, 是], [[a2, a3, s], 30, 0, 30, 35, 是, 是], [[a0, a2, a3, s], 30, 0, 30, 35, 是, 是], [[a1, a2, a3, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, a3, s], 30, 0, 30, 35, 是, 是], [[a4, s], 30, 0, 30, 35, 是, 是], [[a0, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a4, s], 30, 0, 30, 35, 是, 是], [[a2, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a2, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a2, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, a4, s], 30, 0, 30, 35, 是, 是], [[a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a2, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a2, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a2, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, a3, a4, s], 30, 0, 30, 35, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155029,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"],["a0","s"]],"source":"s","sink":"t","capacities":[5,5,6,6,7,7,8,8,9,9,10],"flows":[4,4,5,5,6,6,7,7,8,8,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[10,1],[5,1],[6,1],[7,1],[8,1],[9,1],[5,1],[6,1],[7,1],[8,1],[9,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"flows\":[[0,1],[4,1],[5,1],[6,1],[7,1],[8,1],[4,1],[5,1],[6,1],[7,1],[8,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0030","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[6,6,7,7],\"flows\":[5,5,6,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 11, [[[s], 11, 0, 11, 13, 是, 是], [[a0, s], 11, 0, 11, 13, 是, 是], [[a1, s], 11, 0, 11, 13, 是, 是], [[a0, a1, s], 11, 0, 11, 13, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(11, 11, [[[s], 11, 0, 11, 13, 是, 是], [[a0, s], 11, 0, 11, 13, 是, 是], [[a1, s], 11, 0, 11, 13, 是, 是], [[a0, a1, s], 11, 0, 11, 13, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155030,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[6,6,7,7],"flows":[5,5,6,6],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[6,1],[7,1],[6,1],[7,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[5,1],[6,1],[5,1],[6,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0031","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[7,7,8,8,9,9],\"flows\":[6,6,7,7,8,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 21, [[[s], 21, 0, 21, 24, 是, 是], [[a0, s], 21, 0, 21, 24, 是, 是], [[a1, s], 21, 0, 21, 24, 是, 是], [[a0, a1, s], 21, 0, 21, 24, 是, 是], [[a2, s], 21, 0, 21, 24, 是, 是], [[a0, a2, s], 21, 0, 21, 24, 是, 是], [[a1, a2, s], 21, 0, 21, 24, 是, 是], [[a0, a1, a2, s], 21, 0, 21, 24, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(21, 21, [[[s], 21, 0, 21, 24, 是, 是], [[a0, s], 21, 0, 21, 24, 是, 是], [[a1, s], 21, 0, 21, 24, 是, 是], [[a0, a1, s], 21, 0, 21, 24, 是, 是], [[a2, s], 21, 0, 21, 24, 是, 是], [[a0, a2, s], 21, 0, 21, 24, 是, 是], [[a1, a2, s], 21, 0, 21, 24, 是, 是], [[a0, a1, a2, s], 21, 0, 21, 24, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155031,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[7,7,8,8,9,9],"flows":[6,6,7,7,8,8],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[7,1],[8,1],[9,1],[7,1],[8,1],[9,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"flows\":[[6,1],[7,1],[8,1],[6,1],[7,1],[8,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0032","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[8,8,9,9,10,10,11,11,13],\"flows\":[7,7,8,8,9,9,10,10,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(34, 34, [[[s], 34, 0, 34, 38, 是, 是], [[a0, s], 34, 0, 34, 38, 是, 是], [[a1, s], 34, 0, 34, 38, 是, 是], [[a0, a1, s], 34, 0, 34, 38, 是, 是], [[a2, s], 34, 0, 34, 38, 是, 是], [[a0, a2, s], 34, 0, 34, 38, 是, 是], [[a1, a2, s], 34, 0, 34, 38, 是, 是], [[a0, a1, a2, s], 34, 0, 34, 38, 是, 是], [[a3, s], 34, 0, 34, 38, 是, 是], [[a0, a3, s], 34, 0, 34, 38, 是, 是], [[a1, a3, s], 34, 0, 34, 38, 是, 是], [[a0, a1, a3, s], 34, 0, 34, 38, 是, 是], [[a2, a3, s], 34, 0, 34, 38, 是, 是], [[a0, a2, a3, s], 34, 0, 34, 38, 是, 是], [[a1, a2, a3, s], 34, 0, 34, 38, 是, 是], [[a0, a1, a2, a3, s], 34, 0, 34, 38, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(34, 34, [[[s], 34, 0, 34, 38, 是, 是], [[a0, s], 34, 0, 34, 38, 是, 是], [[a1, s], 34, 0, 34, 38, 是, 是], [[a0, a1, s], 34, 0, 34, 38, 是, 是], [[a2, s], 34, 0, 34, 38, 是, 是], [[a0, a2, s], 34, 0, 34, 38, 是, 是], [[a1, a2, s], 34, 0, 34, 38, 是, 是], [[a0, a1, a2, s], 34, 0, 34, 38, 是, 是], [[a3, s], 34, 0, 34, 38, 是, 是], [[a0, a3, s], 34, 0, 34, 38, 是, 是], [[a1, a3, s], 34, 0, 34, 38, 是, 是], [[a0, a1, a3, s], 34, 0, 34, 38, 是, 是], [[a2, a3, s], 34, 0, 34, 38, 是, 是], [[a0, a2, a3, s], 34, 0, 34, 38, 是, 是], [[a1, a2, a3, s], 34, 0, 34, 38, 是, 是], [[a0, a1, a2, a3, s], 34, 0, 34, 38, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155032,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[8,8,9,9,10,10,11,11,13],"flows":[7,7,8,8,9,9,10,10,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[13,1],[8,1],[9,1],[10,1],[11,1],[8,1],[9,1],[10,1],[11,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"flows\":[[0,1],[7,1],[8,1],[9,1],[10,1],[7,1],[8,1],[9,1],[10,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0033","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[9,9,10,10,11,11,12,12,13,13],\"flows\":[8,8,9,9,10,10,11,11,12,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(50, 50, [[[s], 50, 0, 50, 55, 是, 是], [[a0, s], 50, 0, 50, 55, 是, 是], [[a1, s], 50, 0, 50, 55, 是, 是], [[a0, a1, s], 50, 0, 50, 55, 是, 是], [[a2, s], 50, 0, 50, 55, 是, 是], [[a0, a2, s], 50, 0, 50, 55, 是, 是], [[a1, a2, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, s], 50, 0, 50, 55, 是, 是], [[a3, s], 50, 0, 50, 55, 是, 是], [[a0, a3, s], 50, 0, 50, 55, 是, 是], [[a1, a3, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a3, s], 50, 0, 50, 55, 是, 是], [[a2, a3, s], 50, 0, 50, 55, 是, 是], [[a0, a2, a3, s], 50, 0, 50, 55, 是, 是], [[a1, a2, a3, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, a3, s], 50, 0, 50, 55, 是, 是], [[a4, s], 50, 0, 50, 55, 是, 是], [[a0, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a4, s], 50, 0, 50, 55, 是, 是], [[a2, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a2, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a2, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, a4, s], 50, 0, 50, 55, 是, 是], [[a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a2, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a2, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a2, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, a3, a4, s], 50, 0, 50, 55, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(50, 50, [[[s], 50, 0, 50, 55, 是, 是], [[a0, s], 50, 0, 50, 55, 是, 是], [[a1, s], 50, 0, 50, 55, 是, 是], [[a0, a1, s], 50, 0, 50, 55, 是, 是], [[a2, s], 50, 0, 50, 55, 是, 是], [[a0, a2, s], 50, 0, 50, 55, 是, 是], [[a1, a2, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, s], 50, 0, 50, 55, 是, 是], [[a3, s], 50, 0, 50, 55, 是, 是], [[a0, a3, s], 50, 0, 50, 55, 是, 是], [[a1, a3, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a3, s], 50, 0, 50, 55, 是, 是], [[a2, a3, s], 50, 0, 50, 55, 是, 是], [[a0, a2, a3, s], 50, 0, 50, 55, 是, 是], [[a1, a2, a3, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, a3, s], 50, 0, 50, 55, 是, 是], [[a4, s], 50, 0, 50, 55, 是, 是], [[a0, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a4, s], 50, 0, 50, 55, 是, 是], [[a2, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a2, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a2, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, a4, s], 50, 0, 50, 55, 是, 是], [[a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a2, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a2, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a2, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, a3, a4, s], 50, 0, 50, 55, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155033,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[9,9,10,10,11,11,12,12,13,13],"flows":[8,8,9,9,10,10,11,11,12,12],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[9,1],[10,1],[11,1],[12,1],[13,1],[9,1],[10,1],[11,1],[12,1],[13,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"flows\":[[8,1],[9,1],[10,1],[11,1],[12,1],[8,1],[9,1],[10,1],[11,1],[12,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0034","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[10,10,11,11],\"flows\":[9,9,10,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 19, [[[s], 19, 0, 19, 21, 是, 是], [[a0, s], 19, 0, 19, 21, 是, 是], [[a1, s], 19, 0, 19, 21, 是, 是], [[a0, a1, s], 19, 0, 19, 21, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(19, 19, [[[s], 19, 0, 19, 21, 是, 是], [[a0, s], 19, 0, 19, 21, 是, 是], [[a1, s], 19, 0, 19, 21, 是, 是], [[a0, a1, s], 19, 0, 19, 21, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155034,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[10,10,11,11],"flows":[9,9,10,10],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[10,1],[11,1],[10,1],[11,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[9,1],[10,1],[9,1],[10,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0035","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[11,11,12,12,13,13,16],\"flows\":[10,10,11,11,12,12,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(33, 33, [[[s], 33, 0, 33, 36, 是, 是], [[a0, s], 33, 0, 33, 36, 是, 是], [[a1, s], 33, 0, 33, 36, 是, 是], [[a0, a1, s], 33, 0, 33, 36, 是, 是], [[a2, s], 33, 0, 33, 36, 是, 是], [[a0, a2, s], 33, 0, 33, 36, 是, 是], [[a1, a2, s], 33, 0, 33, 36, 是, 是], [[a0, a1, a2, s], 33, 0, 33, 36, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(33, 33, [[[s], 33, 0, 33, 36, 是, 是], [[a0, s], 33, 0, 33, 36, 是, 是], [[a1, s], 33, 0, 33, 36, 是, 是], [[a0, a1, s], 33, 0, 33, 36, 是, 是], [[a2, s], 33, 0, 33, 36, 是, 是], [[a0, a2, s], 33, 0, 33, 36, 是, 是], [[a1, a2, s], 33, 0, 33, 36, 是, 是], [[a0, a1, a2, s], 33, 0, 33, 36, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155035,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[11,11,12,12,13,13,16],"flows":[10,10,11,11,12,12,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[16,1],[11,1],[12,1],[13,1],[11,1],[12,1],[13,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"flows\":[[0,1],[10,1],[11,1],[12,1],[10,1],[11,1],[12,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0036","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[12,12,13,13,14,14,15,15],\"flows\":[11,11,12,12,13,13,14,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(50, 50, [[[s], 50, 0, 50, 54, 是, 是], [[a0, s], 50, 0, 50, 54, 是, 是], [[a1, s], 50, 0, 50, 54, 是, 是], [[a0, a1, s], 50, 0, 50, 54, 是, 是], [[a2, s], 50, 0, 50, 54, 是, 是], [[a0, a2, s], 50, 0, 50, 54, 是, 是], [[a1, a2, s], 50, 0, 50, 54, 是, 是], [[a0, a1, a2, s], 50, 0, 50, 54, 是, 是], [[a3, s], 50, 0, 50, 54, 是, 是], [[a0, a3, s], 50, 0, 50, 54, 是, 是], [[a1, a3, s], 50, 0, 50, 54, 是, 是], [[a0, a1, a3, s], 50, 0, 50, 54, 是, 是], [[a2, a3, s], 50, 0, 50, 54, 是, 是], [[a0, a2, a3, s], 50, 0, 50, 54, 是, 是], [[a1, a2, a3, s], 50, 0, 50, 54, 是, 是], [[a0, a1, a2, a3, s], 50, 0, 50, 54, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(50, 50, [[[s], 50, 0, 50, 54, 是, 是], [[a0, s], 50, 0, 50, 54, 是, 是], [[a1, s], 50, 0, 50, 54, 是, 是], [[a0, a1, s], 50, 0, 50, 54, 是, 是], [[a2, s], 50, 0, 50, 54, 是, 是], [[a0, a2, s], 50, 0, 50, 54, 是, 是], [[a1, a2, s], 50, 0, 50, 54, 是, 是], [[a0, a1, a2, s], 50, 0, 50, 54, 是, 是], [[a3, s], 50, 0, 50, 54, 是, 是], [[a0, a3, s], 50, 0, 50, 54, 是, 是], [[a1, a3, s], 50, 0, 50, 54, 是, 是], [[a0, a1, a3, s], 50, 0, 50, 54, 是, 是], [[a2, a3, s], 50, 0, 50, 54, 是, 是], [[a0, a2, a3, s], 50, 0, 50, 54, 是, 是], [[a1, a2, a3, s], 50, 0, 50, 54, 是, 是], [[a0, a1, a2, a3, s], 50, 0, 50, 54, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155036,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[12,12,13,13,14,14,15,15],"flows":[11,11,12,12,13,13,14,14],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[12,1],[13,1],[14,1],[15,1],[12,1],[13,1],[14,1],[15,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"flows\":[[11,1],[12,1],[13,1],[14,1],[11,1],[12,1],[13,1],[14,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0037","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[13,13,14,14,15,15,16,16,17,17],\"flows\":[12,12,13,13,14,14,15,15,16,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(70, 70, [[[s], 70, 0, 70, 75, 是, 是], [[a0, s], 70, 0, 70, 75, 是, 是], [[a1, s], 70, 0, 70, 75, 是, 是], [[a0, a1, s], 70, 0, 70, 75, 是, 是], [[a2, s], 70, 0, 70, 75, 是, 是], [[a0, a2, s], 70, 0, 70, 75, 是, 是], [[a1, a2, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, s], 70, 0, 70, 75, 是, 是], [[a3, s], 70, 0, 70, 75, 是, 是], [[a0, a3, s], 70, 0, 70, 75, 是, 是], [[a1, a3, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a3, s], 70, 0, 70, 75, 是, 是], [[a2, a3, s], 70, 0, 70, 75, 是, 是], [[a0, a2, a3, s], 70, 0, 70, 75, 是, 是], [[a1, a2, a3, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, a3, s], 70, 0, 70, 75, 是, 是], [[a4, s], 70, 0, 70, 75, 是, 是], [[a0, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a4, s], 70, 0, 70, 75, 是, 是], [[a2, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a2, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a2, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, a4, s], 70, 0, 70, 75, 是, 是], [[a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a2, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a2, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a2, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, a3, a4, s], 70, 0, 70, 75, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(70, 70, [[[s], 70, 0, 70, 75, 是, 是], [[a0, s], 70, 0, 70, 75, 是, 是], [[a1, s], 70, 0, 70, 75, 是, 是], [[a0, a1, s], 70, 0, 70, 75, 是, 是], [[a2, s], 70, 0, 70, 75, 是, 是], [[a0, a2, s], 70, 0, 70, 75, 是, 是], [[a1, a2, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, s], 70, 0, 70, 75, 是, 是], [[a3, s], 70, 0, 70, 75, 是, 是], [[a0, a3, s], 70, 0, 70, 75, 是, 是], [[a1, a3, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a3, s], 70, 0, 70, 75, 是, 是], [[a2, a3, s], 70, 0, 70, 75, 是, 是], [[a0, a2, a3, s], 70, 0, 70, 75, 是, 是], [[a1, a2, a3, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, a3, s], 70, 0, 70, 75, 是, 是], [[a4, s], 70, 0, 70, 75, 是, 是], [[a0, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a4, s], 70, 0, 70, 75, 是, 是], [[a2, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a2, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a2, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, a4, s], 70, 0, 70, 75, 是, 是], [[a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a2, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a2, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a2, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, a3, a4, s], 70, 0, 70, 75, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155037,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[13,13,14,14,15,15,16,16,17,17],"flows":[12,12,13,13,14,14,15,15,16,16],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[13,1],[14,1],[15,1],[16,1],[17,1],[13,1],[14,1],[15,1],[16,1],[17,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"flows\":[[12,1],[13,1],[14,1],[15,1],[16,1],[12,1],[13,1],[14,1],[15,1],[16,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0038","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[14,14,15,15,19],\"flows\":[13,13,14,14,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(27, 27, [[[s], 27, 0, 27, 29, 是, 是], [[a0, s], 27, 0, 27, 29, 是, 是], [[a1, s], 27, 0, 27, 29, 是, 是], [[a0, a1, s], 27, 0, 27, 29, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(27, 27, [[[s], 27, 0, 27, 29, 是, 是], [[a0, s], 27, 0, 27, 29, 是, 是], [[a1, s], 27, 0, 27, 29, 是, 是], [[a0, a1, s], 27, 0, 27, 29, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155038,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[14,14,15,15,19],"flows":[13,13,14,14,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[19,1],[14,1],[15,1],[14,1],[15,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[0,1],[13,1],[14,1],[13,1],[14,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0039","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[15,15,16,16,17,17],\"flows\":[14,14,15,15,16,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(45, 45, [[[s], 45, 0, 45, 48, 是, 是], [[a0, s], 45, 0, 45, 48, 是, 是], [[a1, s], 45, 0, 45, 48, 是, 是], [[a0, a1, s], 45, 0, 45, 48, 是, 是], [[a2, s], 45, 0, 45, 48, 是, 是], [[a0, a2, s], 45, 0, 45, 48, 是, 是], [[a1, a2, s], 45, 0, 45, 48, 是, 是], [[a0, a1, a2, s], 45, 0, 45, 48, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(45, 45, [[[s], 45, 0, 45, 48, 是, 是], [[a0, s], 45, 0, 45, 48, 是, 是], [[a1, s], 45, 0, 45, 48, 是, 是], [[a0, a1, s], 45, 0, 45, 48, 是, 是], [[a2, s], 45, 0, 45, 48, 是, 是], [[a0, a2, s], 45, 0, 45, 48, 是, 是], [[a1, a2, s], 45, 0, 45, 48, 是, 是], [[a0, a1, a2, s], 45, 0, 45, 48, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155039,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[15,15,16,16,17,17],"flows":[14,14,15,15,16,16],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[15,1],[16,1],[17,1],[15,1],[16,1],[17,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"flows\":[[14,1],[15,1],[16,1],[14,1],[15,1],[16,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0040","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[16,16,17,17,18,18,19,19],\"flows\":[15,15,16,16,17,17,18,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(66, 66, [[[s], 66, 0, 66, 70, 是, 是], [[a0, s], 66, 0, 66, 70, 是, 是], [[a1, s], 66, 0, 66, 70, 是, 是], [[a0, a1, s], 66, 0, 66, 70, 是, 是], [[a2, s], 66, 0, 66, 70, 是, 是], [[a0, a2, s], 66, 0, 66, 70, 是, 是], [[a1, a2, s], 66, 0, 66, 70, 是, 是], [[a0, a1, a2, s], 66, 0, 66, 70, 是, 是], [[a3, s], 66, 0, 66, 70, 是, 是], [[a0, a3, s], 66, 0, 66, 70, 是, 是], [[a1, a3, s], 66, 0, 66, 70, 是, 是], [[a0, a1, a3, s], 66, 0, 66, 70, 是, 是], [[a2, a3, s], 66, 0, 66, 70, 是, 是], [[a0, a2, a3, s], 66, 0, 66, 70, 是, 是], [[a1, a2, a3, s], 66, 0, 66, 70, 是, 是], [[a0, a1, a2, a3, s], 66, 0, 66, 70, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(66, 66, [[[s], 66, 0, 66, 70, 是, 是], [[a0, s], 66, 0, 66, 70, 是, 是], [[a1, s], 66, 0, 66, 70, 是, 是], [[a0, a1, s], 66, 0, 66, 70, 是, 是], [[a2, s], 66, 0, 66, 70, 是, 是], [[a0, a2, s], 66, 0, 66, 70, 是, 是], [[a1, a2, s], 66, 0, 66, 70, 是, 是], [[a0, a1, a2, s], 66, 0, 66, 70, 是, 是], [[a3, s], 66, 0, 66, 70, 是, 是], [[a0, a3, s], 66, 0, 66, 70, 是, 是], [[a1, a3, s], 66, 0, 66, 70, 是, 是], [[a0, a1, a3, s], 66, 0, 66, 70, 是, 是], [[a2, a3, s], 66, 0, 66, 70, 是, 是], [[a0, a2, a3, s], 66, 0, 66, 70, 是, 是], [[a1, a2, a3, s], 66, 0, 66, 70, 是, 是], [[a0, a1, a2, a3, s], 66, 0, 66, 70, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155040,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[16,16,17,17,18,18,19,19],"flows":[15,15,16,16,17,17,18,18],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[16,1],[17,1],[18,1],[19,1],[16,1],[17,1],[18,1],[19,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"flows\":[[15,1],[16,1],[17,1],[18,1],[15,1],[16,1],[17,1],[18,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0041","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[17,17,18,18,19,19,20,20,21,21,22],\"flows\":[16,16,17,17,18,18,19,19,20,20,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(90, 90, [[[s], 90, 0, 90, 95, 是, 是], [[a0, s], 90, 0, 90, 95, 是, 是], [[a1, s], 90, 0, 90, 95, 是, 是], [[a0, a1, s], 90, 0, 90, 95, 是, 是], [[a2, s], 90, 0, 90, 95, 是, 是], [[a0, a2, s], 90, 0, 90, 95, 是, 是], [[a1, a2, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, s], 90, 0, 90, 95, 是, 是], [[a3, s], 90, 0, 90, 95, 是, 是], [[a0, a3, s], 90, 0, 90, 95, 是, 是], [[a1, a3, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a3, s], 90, 0, 90, 95, 是, 是], [[a2, a3, s], 90, 0, 90, 95, 是, 是], [[a0, a2, a3, s], 90, 0, 90, 95, 是, 是], [[a1, a2, a3, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, a3, s], 90, 0, 90, 95, 是, 是], [[a4, s], 90, 0, 90, 95, 是, 是], [[a0, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a4, s], 90, 0, 90, 95, 是, 是], [[a2, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a2, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a2, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, a4, s], 90, 0, 90, 95, 是, 是], [[a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a2, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a2, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a2, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, a3, a4, s], 90, 0, 90, 95, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(90, 90, [[[s], 90, 0, 90, 95, 是, 是], [[a0, s], 90, 0, 90, 95, 是, 是], [[a1, s], 90, 0, 90, 95, 是, 是], [[a0, a1, s], 90, 0, 90, 95, 是, 是], [[a2, s], 90, 0, 90, 95, 是, 是], [[a0, a2, s], 90, 0, 90, 95, 是, 是], [[a1, a2, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, s], 90, 0, 90, 95, 是, 是], [[a3, s], 90, 0, 90, 95, 是, 是], [[a0, a3, s], 90, 0, 90, 95, 是, 是], [[a1, a3, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a3, s], 90, 0, 90, 95, 是, 是], [[a2, a3, s], 90, 0, 90, 95, 是, 是], [[a0, a2, a3, s], 90, 0, 90, 95, 是, 是], [[a1, a2, a3, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, a3, s], 90, 0, 90, 95, 是, 是], [[a4, s], 90, 0, 90, 95, 是, 是], [[a0, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a4, s], 90, 0, 90, 95, 是, 是], [[a2, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a2, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a2, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, a4, s], 90, 0, 90, 95, 是, 是], [[a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a2, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a2, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a2, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, a3, a4, s], 90, 0, 90, 95, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155041,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"],["a0","s"]],"source":"s","sink":"t","capacities":[17,17,18,18,19,19,20,20,21,21,22],"flows":[16,16,17,17,18,18,19,19,20,20,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[22,1],[17,1],[18,1],[19,1],[20,1],[21,1],[17,1],[18,1],[19,1],[20,1],[21,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"flows\":[[0,1],[16,1],[17,1],[18,1],[19,1],[20,1],[16,1],[17,1],[18,1],[19,1],[20,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0042","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[18,18,19,19],\"flows\":[17,17,18,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(35, 35, [[[s], 35, 0, 35, 37, 是, 是], [[a0, s], 35, 0, 35, 37, 是, 是], [[a1, s], 35, 0, 35, 37, 是, 是], [[a0, a1, s], 35, 0, 35, 37, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(35, 35, [[[s], 35, 0, 35, 37, 是, 是], [[a0, s], 35, 0, 35, 37, 是, 是], [[a1, s], 35, 0, 35, 37, 是, 是], [[a0, a1, s], 35, 0, 35, 37, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155042,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[18,18,19,19],"flows":[17,17,18,18],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[18,1],[19,1],[18,1],[19,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[17,1],[18,1],[17,1],[18,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0043","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[19,19,20,20,21,21],\"flows\":[18,18,19,19,20,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(57, 57, [[[s], 57, 0, 57, 60, 是, 是], [[a0, s], 57, 0, 57, 60, 是, 是], [[a1, s], 57, 0, 57, 60, 是, 是], [[a0, a1, s], 57, 0, 57, 60, 是, 是], [[a2, s], 57, 0, 57, 60, 是, 是], [[a0, a2, s], 57, 0, 57, 60, 是, 是], [[a1, a2, s], 57, 0, 57, 60, 是, 是], [[a0, a1, a2, s], 57, 0, 57, 60, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(57, 57, [[[s], 57, 0, 57, 60, 是, 是], [[a0, s], 57, 0, 57, 60, 是, 是], [[a1, s], 57, 0, 57, 60, 是, 是], [[a0, a1, s], 57, 0, 57, 60, 是, 是], [[a2, s], 57, 0, 57, 60, 是, 是], [[a0, a2, s], 57, 0, 57, 60, 是, 是], [[a1, a2, s], 57, 0, 57, 60, 是, 是], [[a0, a1, a2, s], 57, 0, 57, 60, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155043,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[19,19,20,20,21,21],"flows":[18,18,19,19,20,20],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[19,1],[20,1],[21,1],[19,1],[20,1],[21,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"flows\":[[18,1],[19,1],[20,1],[18,1],[19,1],[20,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0044","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[20,20,21,21,22,22,23,23,25],\"flows\":[19,19,20,20,21,21,22,22,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(82, 82, [[[s], 82, 0, 82, 86, 是, 是], [[a0, s], 82, 0, 82, 86, 是, 是], [[a1, s], 82, 0, 82, 86, 是, 是], [[a0, a1, s], 82, 0, 82, 86, 是, 是], [[a2, s], 82, 0, 82, 86, 是, 是], [[a0, a2, s], 82, 0, 82, 86, 是, 是], [[a1, a2, s], 82, 0, 82, 86, 是, 是], [[a0, a1, a2, s], 82, 0, 82, 86, 是, 是], [[a3, s], 82, 0, 82, 86, 是, 是], [[a0, a3, s], 82, 0, 82, 86, 是, 是], [[a1, a3, s], 82, 0, 82, 86, 是, 是], [[a0, a1, a3, s], 82, 0, 82, 86, 是, 是], [[a2, a3, s], 82, 0, 82, 86, 是, 是], [[a0, a2, a3, s], 82, 0, 82, 86, 是, 是], [[a1, a2, a3, s], 82, 0, 82, 86, 是, 是], [[a0, a1, a2, a3, s], 82, 0, 82, 86, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(82, 82, [[[s], 82, 0, 82, 86, 是, 是], [[a0, s], 82, 0, 82, 86, 是, 是], [[a1, s], 82, 0, 82, 86, 是, 是], [[a0, a1, s], 82, 0, 82, 86, 是, 是], [[a2, s], 82, 0, 82, 86, 是, 是], [[a0, a2, s], 82, 0, 82, 86, 是, 是], [[a1, a2, s], 82, 0, 82, 86, 是, 是], [[a0, a1, a2, s], 82, 0, 82, 86, 是, 是], [[a3, s], 82, 0, 82, 86, 是, 是], [[a0, a3, s], 82, 0, 82, 86, 是, 是], [[a1, a3, s], 82, 0, 82, 86, 是, 是], [[a0, a1, a3, s], 82, 0, 82, 86, 是, 是], [[a2, a3, s], 82, 0, 82, 86, 是, 是], [[a0, a2, a3, s], 82, 0, 82, 86, 是, 是], [[a1, a2, a3, s], 82, 0, 82, 86, 是, 是], [[a0, a1, a2, a3, s], 82, 0, 82, 86, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155044,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[20,20,21,21,22,22,23,23,25],"flows":[19,19,20,20,21,21,22,22,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[25,1],[20,1],[21,1],[22,1],[23,1],[20,1],[21,1],[22,1],[23,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"flows\":[[0,1],[19,1],[20,1],[21,1],[22,1],[19,1],[20,1],[21,1],[22,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0045","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[21,21,22,22,23,23,24,24,25,25],\"flows\":[20,20,21,21,22,22,23,23,24,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(110, 110, [[[s], 110, 0, 110, 115, 是, 是], [[a0, s], 110, 0, 110, 115, 是, 是], [[a1, s], 110, 0, 110, 115, 是, 是], [[a0, a1, s], 110, 0, 110, 115, 是, 是], [[a2, s], 110, 0, 110, 115, 是, 是], [[a0, a2, s], 110, 0, 110, 115, 是, 是], [[a1, a2, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, s], 110, 0, 110, 115, 是, 是], [[a3, s], 110, 0, 110, 115, 是, 是], [[a0, a3, s], 110, 0, 110, 115, 是, 是], [[a1, a3, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a3, s], 110, 0, 110, 115, 是, 是], [[a2, a3, s], 110, 0, 110, 115, 是, 是], [[a0, a2, a3, s], 110, 0, 110, 115, 是, 是], [[a1, a2, a3, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, a3, s], 110, 0, 110, 115, 是, 是], [[a4, s], 110, 0, 110, 115, 是, 是], [[a0, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a4, s], 110, 0, 110, 115, 是, 是], [[a2, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a2, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a2, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, a4, s], 110, 0, 110, 115, 是, 是], [[a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a2, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a2, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a2, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, a3, a4, s], 110, 0, 110, 115, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(110, 110, [[[s], 110, 0, 110, 115, 是, 是], [[a0, s], 110, 0, 110, 115, 是, 是], [[a1, s], 110, 0, 110, 115, 是, 是], [[a0, a1, s], 110, 0, 110, 115, 是, 是], [[a2, s], 110, 0, 110, 115, 是, 是], [[a0, a2, s], 110, 0, 110, 115, 是, 是], [[a1, a2, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, s], 110, 0, 110, 115, 是, 是], [[a3, s], 110, 0, 110, 115, 是, 是], [[a0, a3, s], 110, 0, 110, 115, 是, 是], [[a1, a3, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a3, s], 110, 0, 110, 115, 是, 是], [[a2, a3, s], 110, 0, 110, 115, 是, 是], [[a0, a2, a3, s], 110, 0, 110, 115, 是, 是], [[a1, a2, a3, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, a3, s], 110, 0, 110, 115, 是, 是], [[a4, s], 110, 0, 110, 115, 是, 是], [[a0, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a4, s], 110, 0, 110, 115, 是, 是], [[a2, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a2, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a2, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, a4, s], 110, 0, 110, 115, 是, 是], [[a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a2, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a2, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a2, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, a3, a4, s], 110, 0, 110, 115, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155045,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[21,21,22,22,23,23,24,24,25,25],"flows":[20,20,21,21,22,22,23,23,24,24],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[21,1],[22,1],[23,1],[24,1],[25,1],[21,1],[22,1],[23,1],[24,1],[25,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"flows\":[[20,1],[21,1],[22,1],[23,1],[24,1],[20,1],[21,1],[22,1],[23,1],[24,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0046","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[22,22,23,23],\"flows\":[21,21,22,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(43, 43, [[[s], 43, 0, 43, 45, 是, 是], [[a0, s], 43, 0, 43, 45, 是, 是], [[a1, s], 43, 0, 43, 45, 是, 是], [[a0, a1, s], 43, 0, 43, 45, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(43, 43, [[[s], 43, 0, 43, 45, 是, 是], [[a0, s], 43, 0, 43, 45, 是, 是], [[a1, s], 43, 0, 43, 45, 是, 是], [[a0, a1, s], 43, 0, 43, 45, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155046,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[22,22,23,23],"flows":[21,21,22,22],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[22,1],[23,1],[22,1],[23,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[21,1],[22,1],[21,1],[22,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0047","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[23,23,24,24,25,25,28],\"flows\":[22,22,23,23,24,24,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(69, 69, [[[s], 69, 0, 69, 72, 是, 是], [[a0, s], 69, 0, 69, 72, 是, 是], [[a1, s], 69, 0, 69, 72, 是, 是], [[a0, a1, s], 69, 0, 69, 72, 是, 是], [[a2, s], 69, 0, 69, 72, 是, 是], [[a0, a2, s], 69, 0, 69, 72, 是, 是], [[a1, a2, s], 69, 0, 69, 72, 是, 是], [[a0, a1, a2, s], 69, 0, 69, 72, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(69, 69, [[[s], 69, 0, 69, 72, 是, 是], [[a0, s], 69, 0, 69, 72, 是, 是], [[a1, s], 69, 0, 69, 72, 是, 是], [[a0, a1, s], 69, 0, 69, 72, 是, 是], [[a2, s], 69, 0, 69, 72, 是, 是], [[a0, a2, s], 69, 0, 69, 72, 是, 是], [[a1, a2, s], 69, 0, 69, 72, 是, 是], [[a0, a1, a2, s], 69, 0, 69, 72, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155047,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[23,23,24,24,25,25,28],"flows":[22,22,23,23,24,24,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[28,1],[23,1],[24,1],[25,1],[23,1],[24,1],[25,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"flows\":[[0,1],[22,1],[23,1],[24,1],[22,1],[23,1],[24,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0048","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[24,24,25,25,26,26,27,27],\"flows\":[23,23,24,24,25,25,26,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(98, 98, [[[s], 98, 0, 98, 102, 是, 是], [[a0, s], 98, 0, 98, 102, 是, 是], [[a1, s], 98, 0, 98, 102, 是, 是], [[a0, a1, s], 98, 0, 98, 102, 是, 是], [[a2, s], 98, 0, 98, 102, 是, 是], [[a0, a2, s], 98, 0, 98, 102, 是, 是], [[a1, a2, s], 98, 0, 98, 102, 是, 是], [[a0, a1, a2, s], 98, 0, 98, 102, 是, 是], [[a3, s], 98, 0, 98, 102, 是, 是], [[a0, a3, s], 98, 0, 98, 102, 是, 是], [[a1, a3, s], 98, 0, 98, 102, 是, 是], [[a0, a1, a3, s], 98, 0, 98, 102, 是, 是], [[a2, a3, s], 98, 0, 98, 102, 是, 是], [[a0, a2, a3, s], 98, 0, 98, 102, 是, 是], [[a1, a2, a3, s], 98, 0, 98, 102, 是, 是], [[a0, a1, a2, a3, s], 98, 0, 98, 102, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(98, 98, [[[s], 98, 0, 98, 102, 是, 是], [[a0, s], 98, 0, 98, 102, 是, 是], [[a1, s], 98, 0, 98, 102, 是, 是], [[a0, a1, s], 98, 0, 98, 102, 是, 是], [[a2, s], 98, 0, 98, 102, 是, 是], [[a0, a2, s], 98, 0, 98, 102, 是, 是], [[a1, a2, s], 98, 0, 98, 102, 是, 是], [[a0, a1, a2, s], 98, 0, 98, 102, 是, 是], [[a3, s], 98, 0, 98, 102, 是, 是], [[a0, a3, s], 98, 0, 98, 102, 是, 是], [[a1, a3, s], 98, 0, 98, 102, 是, 是], [[a0, a1, a3, s], 98, 0, 98, 102, 是, 是], [[a2, a3, s], 98, 0, 98, 102, 是, 是], [[a0, a2, a3, s], 98, 0, 98, 102, 是, 是], [[a1, a2, a3, s], 98, 0, 98, 102, 是, 是], [[a0, a1, a2, a3, s], 98, 0, 98, 102, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155048,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[24,24,25,25,26,26,27,27],"flows":[23,23,24,24,25,25,26,26],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[24,1],[25,1],[26,1],[27,1],[24,1],[25,1],[26,1],[27,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"flows\":[[23,1],[24,1],[25,1],[26,1],[23,1],[24,1],[25,1],[26,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0049","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[25,25,26,26,27,27,28,28,29,29],\"flows\":[24,24,25,25,26,26,27,27,28,28]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(130, 130, [[[s], 130, 0, 130, 135, 是, 是], [[a0, s], 130, 0, 130, 135, 是, 是], [[a1, s], 130, 0, 130, 135, 是, 是], [[a0, a1, s], 130, 0, 130, 135, 是, 是], [[a2, s], 130, 0, 130, 135, 是, 是], [[a0, a2, s], 130, 0, 130, 135, 是, 是], [[a1, a2, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, s], 130, 0, 130, 135, 是, 是], [[a3, s], 130, 0, 130, 135, 是, 是], [[a0, a3, s], 130, 0, 130, 135, 是, 是], [[a1, a3, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a3, s], 130, 0, 130, 135, 是, 是], [[a2, a3, s], 130, 0, 130, 135, 是, 是], [[a0, a2, a3, s], 130, 0, 130, 135, 是, 是], [[a1, a2, a3, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, a3, s], 130, 0, 130, 135, 是, 是], [[a4, s], 130, 0, 130, 135, 是, 是], [[a0, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a4, s], 130, 0, 130, 135, 是, 是], [[a2, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a2, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a2, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, a4, s], 130, 0, 130, 135, 是, 是], [[a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a2, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a2, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a2, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, a3, a4, s], 130, 0, 130, 135, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(130, 130, [[[s], 130, 0, 130, 135, 是, 是], [[a0, s], 130, 0, 130, 135, 是, 是], [[a1, s], 130, 0, 130, 135, 是, 是], [[a0, a1, s], 130, 0, 130, 135, 是, 是], [[a2, s], 130, 0, 130, 135, 是, 是], [[a0, a2, s], 130, 0, 130, 135, 是, 是], [[a1, a2, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, s], 130, 0, 130, 135, 是, 是], [[a3, s], 130, 0, 130, 135, 是, 是], [[a0, a3, s], 130, 0, 130, 135, 是, 是], [[a1, a3, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a3, s], 130, 0, 130, 135, 是, 是], [[a2, a3, s], 130, 0, 130, 135, 是, 是], [[a0, a2, a3, s], 130, 0, 130, 135, 是, 是], [[a1, a2, a3, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, a3, s], 130, 0, 130, 135, 是, 是], [[a4, s], 130, 0, 130, 135, 是, 是], [[a0, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a4, s], 130, 0, 130, 135, 是, 是], [[a2, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a2, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a2, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, a4, s], 130, 0, 130, 135, 是, 是], [[a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a2, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a2, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a2, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, a3, a4, s], 130, 0, 130, 135, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155049,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[25,25,26,26,27,27,28,28,29,29],"flows":[24,24,25,25,26,26,27,27,28,28],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[25,1],[26,1],[27,1],[28,1],[29,1],[25,1],[26,1],[27,1],[28,1],[29,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"flows\":[[24,1],[25,1],[26,1],[27,1],[28,1],[24,1],[25,1],[26,1],[27,1],[28,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0050","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[26,26,27,27,31],\"flows\":[25,25,26,26,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(51, 51, [[[s], 51, 0, 51, 53, 是, 是], [[a0, s], 51, 0, 51, 53, 是, 是], [[a1, s], 51, 0, 51, 53, 是, 是], [[a0, a1, s], 51, 0, 51, 53, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation_zh":"Wave6第九層核對=(51, 51, [[[s], 51, 0, 51, 53, 是, 是], [[a0, s], 51, 0, 51, 53, 是, 是], [[a1, s], 51, 0, 51, 53, 是, 是], [[a0, a1, s], 51, 0, 51, 53, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155050,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[26,26,27,27,31],"flows":[25,25,26,26,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-flow-cut-upper-bound","semantic_template":"qf-mod46-mfmc-flow-cut-upper-bound/{\"capacities\":[[31,1],[26,1],[27,1],[26,1],[27,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[0,1],[25,1],[26,1],[25,1],[26,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0051","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[2,2,3,3,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 2], [[s, a1], 3]], 5], [[a0, s], [[[a0, t], 2], [[s, a1], 3]], 5], [[a1, s], [[[a1, t], 3], [[s, a0], 2]], 5], [[a0, a1, s], [[[a0, t], 2], [[a1, t], 3]], 5]], 5, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 2], [[s, a1], 3]], 5], [[a0, s], [[[a0, t], 2], [[s, a1], 3]], 5], [[a1, s], [[[a1, t], 3], [[s, a0], 2]], 5], [[a0, a1, s], [[[a0, t], 2], [[a1, t], 3]], 5]], 5, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155051,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[2,2,3,3,7],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[7,1],[2,1],[3,1],[2,1],[3,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0052","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[3,3,4,4,5,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a0, s], [[[a0, t], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a1, s], [[[a1, t], 4], [[s, a0], 3], [[s, a2], 5]], 12], [[a0, a1, s], [[[a0, t], 3], [[a1, t], 4], [[s, a2], 5]], 12], [[a2, s], [[[a2, t], 5], [[s, a0], 3], [[s, a1], 4]], 12], [[a0, a2, s], [[[a0, t], 3], [[a2, t], 5], [[s, a1], 4]], 12], [[a1, a2, s], [[[a1, t], 4], [[a2, t], 5], [[s, a0], 3]], 12], [[a0, a1, a2, s], [[[a0, t], 3], [[a1, t], 4], [[a2, t], 5]], 12]], 12, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a0, s], [[[a0, t], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a1, s], [[[a1, t], 4], [[s, a0], 3], [[s, a2], 5]], 12], [[a0, a1, s], [[[a0, t], 3], [[a1, t], 4], [[s, a2], 5]], 12], [[a2, s], [[[a2, t], 5], [[s, a0], 3], [[s, a1], 4]], 12], [[a0, a2, s], [[[a0, t], 3], [[a2, t], 5], [[s, a1], 4]], 12], [[a1, a2, s], [[[a1, t], 4], [[a2, t], 5], [[s, a0], 3]], 12], [[a0, a1, a2, s], [[[a0, t], 3], [[a1, t], 4], [[a2, t], 5]], 12]], 12, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155052,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[3,3,4,4,5,5],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[3,1],[4,1],[5,1],[3,1],[4,1],[5,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0053","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[4,4,5,5,6,6,7,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, s], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a1, s], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, a1, s], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a2, s], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22], [[a0, a2, s], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22], [[a1, a2, s], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22], [[a0, a1, a2, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22], [[a3, s], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22], [[a0, a3, s], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22], [[a1, a3, s], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22], [[a0, a1, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22], [[a2, a3, s], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22], [[a0, a2, a3, s], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22], [[a1, a2, a3, s], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22], [[a0, a1, a2, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]], 22, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, s], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a1, s], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, a1, s], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a2, s], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22], [[a0, a2, s], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22], [[a1, a2, s], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22], [[a0, a1, a2, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22], [[a3, s], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22], [[a0, a3, s], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22], [[a1, a3, s], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22], [[a0, a1, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22], [[a2, a3, s], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22], [[a0, a2, a3, s], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22], [[a1, a2, a3, s], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22], [[a0, a1, a2, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]], 22, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155053,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[4,4,5,5,6,6,7,7],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[4,1],[5,1],[6,1],[7,1],[4,1],[5,1],[6,1],[7,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0054","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[5,5,6,6,7,7,8,8,9,9,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, s], [[[a0, t], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, s], [[[a1, t], 6], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, s], [[[a0, t], 5], [[a1, t], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a2, s], [[[a2, t], 7], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a2, s], [[[a0, t], 5], [[a2, t], 7], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, a2, s], [[[a1, t], 6], [[a2, t], 7], [[s, a0], 5], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, a2, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a3, s], [[[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a3, s], [[[a0, t], 5], [[a3, t], 8], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a1, a3, s], [[[a1, t], 6], [[a3, t], 8], [[s, a0], 5], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a1, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[s, a2], 7], [[s, a4], 9]], 35], [[a2, a3, s], [[[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a4], 9]], 35], [[a0, a2, a3, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[s, a1], 6], [[s, a4], 9]], 35], [[a1, a2, a3, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a4], 9]], 35], [[a0, a1, a2, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a4], 9]], 35], [[a4, s], [[[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a4, s], [[[a0, t], 5], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a1, a4, s], [[[a1, t], 6], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a1, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a4, t], 9], [[s, a2], 7], [[s, a3], 8]], 35], [[a2, a4, s], [[[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8]], 35], [[a0, a2, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a4, t], 9], [[s, a1], 6], [[s, a3], 8]], 35], [[a1, a2, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a3], 8]], 35], [[a0, a1, a2, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a3], 8]], 35], [[a3, a4, s], [[[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7]], 35], [[a0, a3, a4, s], [[[a0, t], 5], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7]], 35], [[a1, a3, a4, s], [[[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7]], 35], [[a0, a1, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a2], 7]], 35], [[a2, a3, a4, s], [[[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6]], 35], [[a0, a2, a3, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6]], 35], [[a1, a2, a3, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5]], 35], [[a0, a1, a2, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9]], 35]], 35, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, s], [[[a0, t], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, s], [[[a1, t], 6], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, s], [[[a0, t], 5], [[a1, t], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a2, s], [[[a2, t], 7], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a2, s], [[[a0, t], 5], [[a2, t], 7], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, a2, s], [[[a1, t], 6], [[a2, t], 7], [[s, a0], 5], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, a2, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a3, s], [[[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a3, s], [[[a0, t], 5], [[a3, t], 8], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a1, a3, s], [[[a1, t], 6], [[a3, t], 8], [[s, a0], 5], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a1, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[s, a2], 7], [[s, a4], 9]], 35], [[a2, a3, s], [[[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a4], 9]], 35], [[a0, a2, a3, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[s, a1], 6], [[s, a4], 9]], 35], [[a1, a2, a3, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a4], 9]], 35], [[a0, a1, a2, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a4], 9]], 35], [[a4, s], [[[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a4, s], [[[a0, t], 5], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a1, a4, s], [[[a1, t], 6], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a1, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a4, t], 9], [[s, a2], 7], [[s, a3], 8]], 35], [[a2, a4, s], [[[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8]], 35], [[a0, a2, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a4, t], 9], [[s, a1], 6], [[s, a3], 8]], 35], [[a1, a2, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a3], 8]], 35], [[a0, a1, a2, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a3], 8]], 35], [[a3, a4, s], [[[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7]], 35], [[a0, a3, a4, s], [[[a0, t], 5], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7]], 35], [[a1, a3, a4, s], [[[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7]], 35], [[a0, a1, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a2], 7]], 35], [[a2, a3, a4, s], [[[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6]], 35], [[a0, a2, a3, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6]], 35], [[a1, a2, a3, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5]], 35], [[a0, a1, a2, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9]], 35]], 35, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155054,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"],["a0","s"]],"source":"s","sink":"t","capacities":[5,5,6,6,7,7,8,8,9,9,10],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[10,1],[5,1],[6,1],[7,1],[8,1],[9,1],[5,1],[6,1],[7,1],[8,1],[9,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0055","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[6,6,7,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 6], [[s, a1], 7]], 13], [[a0, s], [[[a0, t], 6], [[s, a1], 7]], 13], [[a1, s], [[[a1, t], 7], [[s, a0], 6]], 13], [[a0, a1, s], [[[a0, t], 6], [[a1, t], 7]], 13]], 13, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 6], [[s, a1], 7]], 13], [[a0, s], [[[a0, t], 6], [[s, a1], 7]], 13], [[a1, s], [[[a1, t], 7], [[s, a0], 6]], 13], [[a0, a1, s], [[[a0, t], 6], [[a1, t], 7]], 13]], 13, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155055,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[6,6,7,7],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[6,1],[7,1],[6,1],[7,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0056","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[7,7,8,8,9,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a0, s], [[[a0, t], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a1, s], [[[a1, t], 8], [[s, a0], 7], [[s, a2], 9]], 24], [[a0, a1, s], [[[a0, t], 7], [[a1, t], 8], [[s, a2], 9]], 24], [[a2, s], [[[a2, t], 9], [[s, a0], 7], [[s, a1], 8]], 24], [[a0, a2, s], [[[a0, t], 7], [[a2, t], 9], [[s, a1], 8]], 24], [[a1, a2, s], [[[a1, t], 8], [[a2, t], 9], [[s, a0], 7]], 24], [[a0, a1, a2, s], [[[a0, t], 7], [[a1, t], 8], [[a2, t], 9]], 24]], 24, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a0, s], [[[a0, t], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a1, s], [[[a1, t], 8], [[s, a0], 7], [[s, a2], 9]], 24], [[a0, a1, s], [[[a0, t], 7], [[a1, t], 8], [[s, a2], 9]], 24], [[a2, s], [[[a2, t], 9], [[s, a0], 7], [[s, a1], 8]], 24], [[a0, a2, s], [[[a0, t], 7], [[a2, t], 9], [[s, a1], 8]], 24], [[a1, a2, s], [[[a1, t], 8], [[a2, t], 9], [[s, a0], 7]], 24], [[a0, a1, a2, s], [[[a0, t], 7], [[a1, t], 8], [[a2, t], 9]], 24]], 24, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155056,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[7,7,8,8,9,9],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[7,1],[8,1],[9,1],[7,1],[8,1],[9,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0057","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[8,8,9,9,10,10,11,11,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, s], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a1, s], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, a1, s], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a2, s], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38], [[a0, a2, s], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38], [[a1, a2, s], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38], [[a0, a1, a2, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38], [[a3, s], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38], [[a0, a3, s], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38], [[a1, a3, s], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38], [[a0, a1, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38], [[a2, a3, s], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38], [[a0, a2, a3, s], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38], [[a1, a2, a3, s], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38], [[a0, a1, a2, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]], 38, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, s], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a1, s], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, a1, s], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a2, s], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38], [[a0, a2, s], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38], [[a1, a2, s], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38], [[a0, a1, a2, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38], [[a3, s], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38], [[a0, a3, s], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38], [[a1, a3, s], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38], [[a0, a1, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38], [[a2, a3, s], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38], [[a0, a2, a3, s], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38], [[a1, a2, a3, s], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38], [[a0, a1, a2, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]], 38, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155057,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[8,8,9,9,10,10,11,11,13],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[13,1],[8,1],[9,1],[10,1],[11,1],[8,1],[9,1],[10,1],[11,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0058","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[9,9,10,10,11,11,12,12,13,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 55, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 55, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155058,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[9,9,10,10,11,11,12,12,13,13],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[9,1],[10,1],[11,1],[12,1],[13,1],[9,1],[10,1],[11,1],[12,1],[13,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0059","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[10,10,11,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 10], [[s, a1], 11]], 21], [[a0, s], [[[a0, t], 10], [[s, a1], 11]], 21], [[a1, s], [[[a1, t], 11], [[s, a0], 10]], 21], [[a0, a1, s], [[[a0, t], 10], [[a1, t], 11]], 21]], 21, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 10], [[s, a1], 11]], 21], [[a0, s], [[[a0, t], 10], [[s, a1], 11]], 21], [[a1, s], [[[a1, t], 11], [[s, a0], 10]], 21], [[a0, a1, s], [[[a0, t], 10], [[a1, t], 11]], 21]], 21, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155059,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[10,10,11,11],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[10,1],[11,1],[10,1],[11,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0060","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[11,11,12,12,13,13,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a0, s], [[[a0, t], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a1, s], [[[a1, t], 12], [[s, a0], 11], [[s, a2], 13]], 36], [[a0, a1, s], [[[a0, t], 11], [[a1, t], 12], [[s, a2], 13]], 36], [[a2, s], [[[a2, t], 13], [[s, a0], 11], [[s, a1], 12]], 36], [[a0, a2, s], [[[a0, t], 11], [[a2, t], 13], [[s, a1], 12]], 36], [[a1, a2, s], [[[a1, t], 12], [[a2, t], 13], [[s, a0], 11]], 36], [[a0, a1, a2, s], [[[a0, t], 11], [[a1, t], 12], [[a2, t], 13]], 36]], 36, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a0, s], [[[a0, t], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a1, s], [[[a1, t], 12], [[s, a0], 11], [[s, a2], 13]], 36], [[a0, a1, s], [[[a0, t], 11], [[a1, t], 12], [[s, a2], 13]], 36], [[a2, s], [[[a2, t], 13], [[s, a0], 11], [[s, a1], 12]], 36], [[a0, a2, s], [[[a0, t], 11], [[a2, t], 13], [[s, a1], 12]], 36], [[a1, a2, s], [[[a1, t], 12], [[a2, t], 13], [[s, a0], 11]], 36], [[a0, a1, a2, s], [[[a0, t], 11], [[a1, t], 12], [[a2, t], 13]], 36]], 36, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155060,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[11,11,12,12,13,13,16],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[16,1],[11,1],[12,1],[13,1],[11,1],[12,1],[13,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0061","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[12,12,13,13,14,14,15,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, s], [[[a0, t], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a1, s], [[[a1, t], 13], [[s, a0], 12], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, a1, s], [[[a0, t], 12], [[a1, t], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a2, s], [[[a2, t], 14], [[s, a0], 12], [[s, a1], 13], [[s, a3], 15]], 54], [[a0, a2, s], [[[a0, t], 12], [[a2, t], 14], [[s, a1], 13], [[s, a3], 15]], 54], [[a1, a2, s], [[[a1, t], 13], [[a2, t], 14], [[s, a0], 12], [[s, a3], 15]], 54], [[a0, a1, a2, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[s, a3], 15]], 54], [[a3, s], [[[a3, t], 15], [[s, a0], 12], [[s, a1], 13], [[s, a2], 14]], 54], [[a0, a3, s], [[[a0, t], 12], [[a3, t], 15], [[s, a1], 13], [[s, a2], 14]], 54], [[a1, a3, s], [[[a1, t], 13], [[a3, t], 15], [[s, a0], 12], [[s, a2], 14]], 54], [[a0, a1, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a3, t], 15], [[s, a2], 14]], 54], [[a2, a3, s], [[[a2, t], 14], [[a3, t], 15], [[s, a0], 12], [[s, a1], 13]], 54], [[a0, a2, a3, s], [[[a0, t], 12], [[a2, t], 14], [[a3, t], 15], [[s, a1], 13]], 54], [[a1, a2, a3, s], [[[a1, t], 13], [[a2, t], 14], [[a3, t], 15], [[s, a0], 12]], 54], [[a0, a1, a2, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[a3, t], 15]], 54]], 54, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, s], [[[a0, t], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a1, s], [[[a1, t], 13], [[s, a0], 12], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, a1, s], [[[a0, t], 12], [[a1, t], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a2, s], [[[a2, t], 14], [[s, a0], 12], [[s, a1], 13], [[s, a3], 15]], 54], [[a0, a2, s], [[[a0, t], 12], [[a2, t], 14], [[s, a1], 13], [[s, a3], 15]], 54], [[a1, a2, s], [[[a1, t], 13], [[a2, t], 14], [[s, a0], 12], [[s, a3], 15]], 54], [[a0, a1, a2, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[s, a3], 15]], 54], [[a3, s], [[[a3, t], 15], [[s, a0], 12], [[s, a1], 13], [[s, a2], 14]], 54], [[a0, a3, s], [[[a0, t], 12], [[a3, t], 15], [[s, a1], 13], [[s, a2], 14]], 54], [[a1, a3, s], [[[a1, t], 13], [[a3, t], 15], [[s, a0], 12], [[s, a2], 14]], 54], [[a0, a1, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a3, t], 15], [[s, a2], 14]], 54], [[a2, a3, s], [[[a2, t], 14], [[a3, t], 15], [[s, a0], 12], [[s, a1], 13]], 54], [[a0, a2, a3, s], [[[a0, t], 12], [[a2, t], 14], [[a3, t], 15], [[s, a1], 13]], 54], [[a1, a2, a3, s], [[[a1, t], 13], [[a2, t], 14], [[a3, t], 15], [[s, a0], 12]], 54], [[a0, a1, a2, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[a3, t], 15]], 54]], 54, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155061,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[12,12,13,13,14,14,15,15],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[12,1],[13,1],[14,1],[15,1],[12,1],[13,1],[14,1],[15,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0062","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[13,13,14,14,15,15,16,16,17,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, s], [[[a0, t], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, s], [[[a1, t], 14], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, s], [[[a0, t], 13], [[a1, t], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a2, s], [[[a2, t], 15], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a2, s], [[[a0, t], 13], [[a2, t], 15], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, a2, s], [[[a1, t], 14], [[a2, t], 15], [[s, a0], 13], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, a2, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a3, s], [[[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a3, s], [[[a0, t], 13], [[a3, t], 16], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a1, a3, s], [[[a1, t], 14], [[a3, t], 16], [[s, a0], 13], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a1, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[s, a2], 15], [[s, a4], 17]], 75], [[a2, a3, s], [[[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a4], 17]], 75], [[a0, a2, a3, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[s, a1], 14], [[s, a4], 17]], 75], [[a1, a2, a3, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a4], 17]], 75], [[a0, a1, a2, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a4], 17]], 75], [[a4, s], [[[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a4, s], [[[a0, t], 13], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a1, a4, s], [[[a1, t], 14], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a1, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a4, t], 17], [[s, a2], 15], [[s, a3], 16]], 75], [[a2, a4, s], [[[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16]], 75], [[a0, a2, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a4, t], 17], [[s, a1], 14], [[s, a3], 16]], 75], [[a1, a2, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a3], 16]], 75], [[a0, a1, a2, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a3], 16]], 75], [[a3, a4, s], [[[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15]], 75], [[a0, a3, a4, s], [[[a0, t], 13], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15]], 75], [[a1, a3, a4, s], [[[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15]], 75], [[a0, a1, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a2], 15]], 75], [[a2, a3, a4, s], [[[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14]], 75], [[a0, a2, a3, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14]], 75], [[a1, a2, a3, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13]], 75], [[a0, a1, a2, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17]], 75]], 75, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, s], [[[a0, t], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, s], [[[a1, t], 14], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, s], [[[a0, t], 13], [[a1, t], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a2, s], [[[a2, t], 15], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a2, s], [[[a0, t], 13], [[a2, t], 15], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, a2, s], [[[a1, t], 14], [[a2, t], 15], [[s, a0], 13], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, a2, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a3, s], [[[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a3, s], [[[a0, t], 13], [[a3, t], 16], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a1, a3, s], [[[a1, t], 14], [[a3, t], 16], [[s, a0], 13], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a1, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[s, a2], 15], [[s, a4], 17]], 75], [[a2, a3, s], [[[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a4], 17]], 75], [[a0, a2, a3, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[s, a1], 14], [[s, a4], 17]], 75], [[a1, a2, a3, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a4], 17]], 75], [[a0, a1, a2, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a4], 17]], 75], [[a4, s], [[[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a4, s], [[[a0, t], 13], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a1, a4, s], [[[a1, t], 14], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a1, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a4, t], 17], [[s, a2], 15], [[s, a3], 16]], 75], [[a2, a4, s], [[[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16]], 75], [[a0, a2, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a4, t], 17], [[s, a1], 14], [[s, a3], 16]], 75], [[a1, a2, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a3], 16]], 75], [[a0, a1, a2, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a3], 16]], 75], [[a3, a4, s], [[[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15]], 75], [[a0, a3, a4, s], [[[a0, t], 13], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15]], 75], [[a1, a3, a4, s], [[[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15]], 75], [[a0, a1, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a2], 15]], 75], [[a2, a3, a4, s], [[[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14]], 75], [[a0, a2, a3, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14]], 75], [[a1, a2, a3, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13]], 75], [[a0, a1, a2, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17]], 75]], 75, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155062,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[13,13,14,14,15,15,16,16,17,17],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[13,1],[14,1],[15,1],[16,1],[17,1],[13,1],[14,1],[15,1],[16,1],[17,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0063","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[14,14,15,15,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 14], [[s, a1], 15]], 29], [[a0, s], [[[a0, t], 14], [[s, a1], 15]], 29], [[a1, s], [[[a1, t], 15], [[s, a0], 14]], 29], [[a0, a1, s], [[[a0, t], 14], [[a1, t], 15]], 29]], 29, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 14], [[s, a1], 15]], 29], [[a0, s], [[[a0, t], 14], [[s, a1], 15]], 29], [[a1, s], [[[a1, t], 15], [[s, a0], 14]], 29], [[a0, a1, s], [[[a0, t], 14], [[a1, t], 15]], 29]], 29, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155063,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[14,14,15,15,19],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[19,1],[14,1],[15,1],[14,1],[15,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0064","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[15,15,16,16,17,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a0, s], [[[a0, t], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a1, s], [[[a1, t], 16], [[s, a0], 15], [[s, a2], 17]], 48], [[a0, a1, s], [[[a0, t], 15], [[a1, t], 16], [[s, a2], 17]], 48], [[a2, s], [[[a2, t], 17], [[s, a0], 15], [[s, a1], 16]], 48], [[a0, a2, s], [[[a0, t], 15], [[a2, t], 17], [[s, a1], 16]], 48], [[a1, a2, s], [[[a1, t], 16], [[a2, t], 17], [[s, a0], 15]], 48], [[a0, a1, a2, s], [[[a0, t], 15], [[a1, t], 16], [[a2, t], 17]], 48]], 48, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a0, s], [[[a0, t], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a1, s], [[[a1, t], 16], [[s, a0], 15], [[s, a2], 17]], 48], [[a0, a1, s], [[[a0, t], 15], [[a1, t], 16], [[s, a2], 17]], 48], [[a2, s], [[[a2, t], 17], [[s, a0], 15], [[s, a1], 16]], 48], [[a0, a2, s], [[[a0, t], 15], [[a2, t], 17], [[s, a1], 16]], 48], [[a1, a2, s], [[[a1, t], 16], [[a2, t], 17], [[s, a0], 15]], 48], [[a0, a1, a2, s], [[[a0, t], 15], [[a1, t], 16], [[a2, t], 17]], 48]], 48, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155064,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[15,15,16,16,17,17],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[15,1],[16,1],[17,1],[15,1],[16,1],[17,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0065","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[16,16,17,17,18,18,19,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, s], [[[a0, t], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a1, s], [[[a1, t], 17], [[s, a0], 16], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, a1, s], [[[a0, t], 16], [[a1, t], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a2, s], [[[a2, t], 18], [[s, a0], 16], [[s, a1], 17], [[s, a3], 19]], 70], [[a0, a2, s], [[[a0, t], 16], [[a2, t], 18], [[s, a1], 17], [[s, a3], 19]], 70], [[a1, a2, s], [[[a1, t], 17], [[a2, t], 18], [[s, a0], 16], [[s, a3], 19]], 70], [[a0, a1, a2, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[s, a3], 19]], 70], [[a3, s], [[[a3, t], 19], [[s, a0], 16], [[s, a1], 17], [[s, a2], 18]], 70], [[a0, a3, s], [[[a0, t], 16], [[a3, t], 19], [[s, a1], 17], [[s, a2], 18]], 70], [[a1, a3, s], [[[a1, t], 17], [[a3, t], 19], [[s, a0], 16], [[s, a2], 18]], 70], [[a0, a1, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a3, t], 19], [[s, a2], 18]], 70], [[a2, a3, s], [[[a2, t], 18], [[a3, t], 19], [[s, a0], 16], [[s, a1], 17]], 70], [[a0, a2, a3, s], [[[a0, t], 16], [[a2, t], 18], [[a3, t], 19], [[s, a1], 17]], 70], [[a1, a2, a3, s], [[[a1, t], 17], [[a2, t], 18], [[a3, t], 19], [[s, a0], 16]], 70], [[a0, a1, a2, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[a3, t], 19]], 70]], 70, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, s], [[[a0, t], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a1, s], [[[a1, t], 17], [[s, a0], 16], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, a1, s], [[[a0, t], 16], [[a1, t], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a2, s], [[[a2, t], 18], [[s, a0], 16], [[s, a1], 17], [[s, a3], 19]], 70], [[a0, a2, s], [[[a0, t], 16], [[a2, t], 18], [[s, a1], 17], [[s, a3], 19]], 70], [[a1, a2, s], [[[a1, t], 17], [[a2, t], 18], [[s, a0], 16], [[s, a3], 19]], 70], [[a0, a1, a2, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[s, a3], 19]], 70], [[a3, s], [[[a3, t], 19], [[s, a0], 16], [[s, a1], 17], [[s, a2], 18]], 70], [[a0, a3, s], [[[a0, t], 16], [[a3, t], 19], [[s, a1], 17], [[s, a2], 18]], 70], [[a1, a3, s], [[[a1, t], 17], [[a3, t], 19], [[s, a0], 16], [[s, a2], 18]], 70], [[a0, a1, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a3, t], 19], [[s, a2], 18]], 70], [[a2, a3, s], [[[a2, t], 18], [[a3, t], 19], [[s, a0], 16], [[s, a1], 17]], 70], [[a0, a2, a3, s], [[[a0, t], 16], [[a2, t], 18], [[a3, t], 19], [[s, a1], 17]], 70], [[a1, a2, a3, s], [[[a1, t], 17], [[a2, t], 18], [[a3, t], 19], [[s, a0], 16]], 70], [[a0, a1, a2, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[a3, t], 19]], 70]], 70, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155065,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[16,16,17,17,18,18,19,19],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[16,1],[17,1],[18,1],[19,1],[16,1],[17,1],[18,1],[19,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0066","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[17,17,18,18,19,19,20,20,21,21,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, s], [[[a0, t], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, s], [[[a1, t], 18], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, s], [[[a0, t], 17], [[a1, t], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a2, s], [[[a2, t], 19], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a2, s], [[[a0, t], 17], [[a2, t], 19], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, a2, s], [[[a1, t], 18], [[a2, t], 19], [[s, a0], 17], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, a2, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a3, s], [[[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a3, s], [[[a0, t], 17], [[a3, t], 20], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a1, a3, s], [[[a1, t], 18], [[a3, t], 20], [[s, a0], 17], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a1, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[s, a2], 19], [[s, a4], 21]], 95], [[a2, a3, s], [[[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a4], 21]], 95], [[a0, a2, a3, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[s, a1], 18], [[s, a4], 21]], 95], [[a1, a2, a3, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a4], 21]], 95], [[a0, a1, a2, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a4], 21]], 95], [[a4, s], [[[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a4, s], [[[a0, t], 17], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a1, a4, s], [[[a1, t], 18], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a1, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a4, t], 21], [[s, a2], 19], [[s, a3], 20]], 95], [[a2, a4, s], [[[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20]], 95], [[a0, a2, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a4, t], 21], [[s, a1], 18], [[s, a3], 20]], 95], [[a1, a2, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a3], 20]], 95], [[a0, a1, a2, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a3], 20]], 95], [[a3, a4, s], [[[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19]], 95], [[a0, a3, a4, s], [[[a0, t], 17], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19]], 95], [[a1, a3, a4, s], [[[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19]], 95], [[a0, a1, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a2], 19]], 95], [[a2, a3, a4, s], [[[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18]], 95], [[a0, a2, a3, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18]], 95], [[a1, a2, a3, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17]], 95], [[a0, a1, a2, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21]], 95]], 95, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, s], [[[a0, t], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, s], [[[a1, t], 18], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, s], [[[a0, t], 17], [[a1, t], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a2, s], [[[a2, t], 19], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a2, s], [[[a0, t], 17], [[a2, t], 19], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, a2, s], [[[a1, t], 18], [[a2, t], 19], [[s, a0], 17], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, a2, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a3, s], [[[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a3, s], [[[a0, t], 17], [[a3, t], 20], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a1, a3, s], [[[a1, t], 18], [[a3, t], 20], [[s, a0], 17], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a1, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[s, a2], 19], [[s, a4], 21]], 95], [[a2, a3, s], [[[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a4], 21]], 95], [[a0, a2, a3, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[s, a1], 18], [[s, a4], 21]], 95], [[a1, a2, a3, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a4], 21]], 95], [[a0, a1, a2, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a4], 21]], 95], [[a4, s], [[[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a4, s], [[[a0, t], 17], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a1, a4, s], [[[a1, t], 18], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a1, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a4, t], 21], [[s, a2], 19], [[s, a3], 20]], 95], [[a2, a4, s], [[[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20]], 95], [[a0, a2, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a4, t], 21], [[s, a1], 18], [[s, a3], 20]], 95], [[a1, a2, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a3], 20]], 95], [[a0, a1, a2, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a3], 20]], 95], [[a3, a4, s], [[[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19]], 95], [[a0, a3, a4, s], [[[a0, t], 17], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19]], 95], [[a1, a3, a4, s], [[[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19]], 95], [[a0, a1, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a2], 19]], 95], [[a2, a3, a4, s], [[[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18]], 95], [[a0, a2, a3, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18]], 95], [[a1, a2, a3, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17]], 95], [[a0, a1, a2, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21]], 95]], 95, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155066,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"],["a0","s"]],"source":"s","sink":"t","capacities":[17,17,18,18,19,19,20,20,21,21,22],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[22,1],[17,1],[18,1],[19,1],[20,1],[21,1],[17,1],[18,1],[19,1],[20,1],[21,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0067","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[18,18,19,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 18], [[s, a1], 19]], 37], [[a0, s], [[[a0, t], 18], [[s, a1], 19]], 37], [[a1, s], [[[a1, t], 19], [[s, a0], 18]], 37], [[a0, a1, s], [[[a0, t], 18], [[a1, t], 19]], 37]], 37, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 18], [[s, a1], 19]], 37], [[a0, s], [[[a0, t], 18], [[s, a1], 19]], 37], [[a1, s], [[[a1, t], 19], [[s, a0], 18]], 37], [[a0, a1, s], [[[a0, t], 18], [[a1, t], 19]], 37]], 37, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155067,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[18,18,19,19],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[18,1],[19,1],[18,1],[19,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0068","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[19,19,20,20,21,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a0, s], [[[a0, t], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a1, s], [[[a1, t], 20], [[s, a0], 19], [[s, a2], 21]], 60], [[a0, a1, s], [[[a0, t], 19], [[a1, t], 20], [[s, a2], 21]], 60], [[a2, s], [[[a2, t], 21], [[s, a0], 19], [[s, a1], 20]], 60], [[a0, a2, s], [[[a0, t], 19], [[a2, t], 21], [[s, a1], 20]], 60], [[a1, a2, s], [[[a1, t], 20], [[a2, t], 21], [[s, a0], 19]], 60], [[a0, a1, a2, s], [[[a0, t], 19], [[a1, t], 20], [[a2, t], 21]], 60]], 60, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a0, s], [[[a0, t], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a1, s], [[[a1, t], 20], [[s, a0], 19], [[s, a2], 21]], 60], [[a0, a1, s], [[[a0, t], 19], [[a1, t], 20], [[s, a2], 21]], 60], [[a2, s], [[[a2, t], 21], [[s, a0], 19], [[s, a1], 20]], 60], [[a0, a2, s], [[[a0, t], 19], [[a2, t], 21], [[s, a1], 20]], 60], [[a1, a2, s], [[[a1, t], 20], [[a2, t], 21], [[s, a0], 19]], 60], [[a0, a1, a2, s], [[[a0, t], 19], [[a1, t], 20], [[a2, t], 21]], 60]], 60, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155068,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[19,19,20,20,21,21],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[19,1],[20,1],[21,1],[19,1],[20,1],[21,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0069","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[20,20,21,21,22,22,23,23,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, s], [[[a0, t], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a1, s], [[[a1, t], 21], [[s, a0], 20], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, a1, s], [[[a0, t], 20], [[a1, t], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a2, s], [[[a2, t], 22], [[s, a0], 20], [[s, a1], 21], [[s, a3], 23]], 86], [[a0, a2, s], [[[a0, t], 20], [[a2, t], 22], [[s, a1], 21], [[s, a3], 23]], 86], [[a1, a2, s], [[[a1, t], 21], [[a2, t], 22], [[s, a0], 20], [[s, a3], 23]], 86], [[a0, a1, a2, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[s, a3], 23]], 86], [[a3, s], [[[a3, t], 23], [[s, a0], 20], [[s, a1], 21], [[s, a2], 22]], 86], [[a0, a3, s], [[[a0, t], 20], [[a3, t], 23], [[s, a1], 21], [[s, a2], 22]], 86], [[a1, a3, s], [[[a1, t], 21], [[a3, t], 23], [[s, a0], 20], [[s, a2], 22]], 86], [[a0, a1, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a3, t], 23], [[s, a2], 22]], 86], [[a2, a3, s], [[[a2, t], 22], [[a3, t], 23], [[s, a0], 20], [[s, a1], 21]], 86], [[a0, a2, a3, s], [[[a0, t], 20], [[a2, t], 22], [[a3, t], 23], [[s, a1], 21]], 86], [[a1, a2, a3, s], [[[a1, t], 21], [[a2, t], 22], [[a3, t], 23], [[s, a0], 20]], 86], [[a0, a1, a2, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[a3, t], 23]], 86]], 86, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, s], [[[a0, t], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a1, s], [[[a1, t], 21], [[s, a0], 20], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, a1, s], [[[a0, t], 20], [[a1, t], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a2, s], [[[a2, t], 22], [[s, a0], 20], [[s, a1], 21], [[s, a3], 23]], 86], [[a0, a2, s], [[[a0, t], 20], [[a2, t], 22], [[s, a1], 21], [[s, a3], 23]], 86], [[a1, a2, s], [[[a1, t], 21], [[a2, t], 22], [[s, a0], 20], [[s, a3], 23]], 86], [[a0, a1, a2, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[s, a3], 23]], 86], [[a3, s], [[[a3, t], 23], [[s, a0], 20], [[s, a1], 21], [[s, a2], 22]], 86], [[a0, a3, s], [[[a0, t], 20], [[a3, t], 23], [[s, a1], 21], [[s, a2], 22]], 86], [[a1, a3, s], [[[a1, t], 21], [[a3, t], 23], [[s, a0], 20], [[s, a2], 22]], 86], [[a0, a1, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a3, t], 23], [[s, a2], 22]], 86], [[a2, a3, s], [[[a2, t], 22], [[a3, t], 23], [[s, a0], 20], [[s, a1], 21]], 86], [[a0, a2, a3, s], [[[a0, t], 20], [[a2, t], 22], [[a3, t], 23], [[s, a1], 21]], 86], [[a1, a2, a3, s], [[[a1, t], 21], [[a2, t], 22], [[a3, t], 23], [[s, a0], 20]], 86], [[a0, a1, a2, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[a3, t], 23]], 86]], 86, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155069,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[20,20,21,21,22,22,23,23,25],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[25,1],[20,1],[21,1],[22,1],[23,1],[20,1],[21,1],[22,1],[23,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0070","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[21,21,22,22,23,23,24,24,25,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, s], [[[a0, t], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, s], [[[a1, t], 22], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, s], [[[a0, t], 21], [[a1, t], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a2, s], [[[a2, t], 23], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a2, s], [[[a0, t], 21], [[a2, t], 23], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, a2, s], [[[a1, t], 22], [[a2, t], 23], [[s, a0], 21], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, a2, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a3, s], [[[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a3, s], [[[a0, t], 21], [[a3, t], 24], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a1, a3, s], [[[a1, t], 22], [[a3, t], 24], [[s, a0], 21], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a1, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[s, a2], 23], [[s, a4], 25]], 115], [[a2, a3, s], [[[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a4], 25]], 115], [[a0, a2, a3, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[s, a1], 22], [[s, a4], 25]], 115], [[a1, a2, a3, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a4], 25]], 115], [[a0, a1, a2, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a4], 25]], 115], [[a4, s], [[[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a4, s], [[[a0, t], 21], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a1, a4, s], [[[a1, t], 22], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a1, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a4, t], 25], [[s, a2], 23], [[s, a3], 24]], 115], [[a2, a4, s], [[[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24]], 115], [[a0, a2, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a4, t], 25], [[s, a1], 22], [[s, a3], 24]], 115], [[a1, a2, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a3], 24]], 115], [[a0, a1, a2, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a3], 24]], 115], [[a3, a4, s], [[[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23]], 115], [[a0, a3, a4, s], [[[a0, t], 21], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23]], 115], [[a1, a3, a4, s], [[[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23]], 115], [[a0, a1, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a2], 23]], 115], [[a2, a3, a4, s], [[[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22]], 115], [[a0, a2, a3, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22]], 115], [[a1, a2, a3, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21]], 115], [[a0, a1, a2, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25]], 115]], 115, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, s], [[[a0, t], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, s], [[[a1, t], 22], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, s], [[[a0, t], 21], [[a1, t], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a2, s], [[[a2, t], 23], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a2, s], [[[a0, t], 21], [[a2, t], 23], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, a2, s], [[[a1, t], 22], [[a2, t], 23], [[s, a0], 21], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, a2, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a3, s], [[[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a3, s], [[[a0, t], 21], [[a3, t], 24], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a1, a3, s], [[[a1, t], 22], [[a3, t], 24], [[s, a0], 21], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a1, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[s, a2], 23], [[s, a4], 25]], 115], [[a2, a3, s], [[[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a4], 25]], 115], [[a0, a2, a3, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[s, a1], 22], [[s, a4], 25]], 115], [[a1, a2, a3, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a4], 25]], 115], [[a0, a1, a2, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a4], 25]], 115], [[a4, s], [[[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a4, s], [[[a0, t], 21], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a1, a4, s], [[[a1, t], 22], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a1, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a4, t], 25], [[s, a2], 23], [[s, a3], 24]], 115], [[a2, a4, s], [[[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24]], 115], [[a0, a2, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a4, t], 25], [[s, a1], 22], [[s, a3], 24]], 115], [[a1, a2, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a3], 24]], 115], [[a0, a1, a2, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a3], 24]], 115], [[a3, a4, s], [[[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23]], 115], [[a0, a3, a4, s], [[[a0, t], 21], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23]], 115], [[a1, a3, a4, s], [[[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23]], 115], [[a0, a1, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a2], 23]], 115], [[a2, a3, a4, s], [[[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22]], 115], [[a0, a2, a3, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22]], 115], [[a1, a2, a3, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21]], 115], [[a0, a1, a2, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25]], 115]], 115, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155070,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[21,21,22,22,23,23,24,24,25,25],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[21,1],[22,1],[23,1],[24,1],[25,1],[21,1],[22,1],[23,1],[24,1],[25,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0071","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[22,22,23,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 22], [[s, a1], 23]], 45], [[a0, s], [[[a0, t], 22], [[s, a1], 23]], 45], [[a1, s], [[[a1, t], 23], [[s, a0], 22]], 45], [[a0, a1, s], [[[a0, t], 22], [[a1, t], 23]], 45]], 45, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 22], [[s, a1], 23]], 45], [[a0, s], [[[a0, t], 22], [[s, a1], 23]], 45], [[a1, s], [[[a1, t], 23], [[s, a0], 22]], 45], [[a0, a1, s], [[[a0, t], 22], [[a1, t], 23]], 45]], 45, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155071,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[22,22,23,23],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[22,1],[23,1],[22,1],[23,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0072","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[23,23,24,24,25,25,28]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a0, s], [[[a0, t], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a1, s], [[[a1, t], 24], [[s, a0], 23], [[s, a2], 25]], 72], [[a0, a1, s], [[[a0, t], 23], [[a1, t], 24], [[s, a2], 25]], 72], [[a2, s], [[[a2, t], 25], [[s, a0], 23], [[s, a1], 24]], 72], [[a0, a2, s], [[[a0, t], 23], [[a2, t], 25], [[s, a1], 24]], 72], [[a1, a2, s], [[[a1, t], 24], [[a2, t], 25], [[s, a0], 23]], 72], [[a0, a1, a2, s], [[[a0, t], 23], [[a1, t], 24], [[a2, t], 25]], 72]], 72, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a0, s], [[[a0, t], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a1, s], [[[a1, t], 24], [[s, a0], 23], [[s, a2], 25]], 72], [[a0, a1, s], [[[a0, t], 23], [[a1, t], 24], [[s, a2], 25]], 72], [[a2, s], [[[a2, t], 25], [[s, a0], 23], [[s, a1], 24]], 72], [[a0, a2, s], [[[a0, t], 23], [[a2, t], 25], [[s, a1], 24]], 72], [[a1, a2, s], [[[a1, t], 24], [[a2, t], 25], [[s, a0], 23]], 72], [[a0, a1, a2, s], [[[a0, t], 23], [[a1, t], 24], [[a2, t], 25]], 72]], 72, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155072,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[23,23,24,24,25,25,28],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[28,1],[23,1],[24,1],[25,1],[23,1],[24,1],[25,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0073","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[24,24,25,25,26,26,27,27]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, s], [[[a0, t], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a1, s], [[[a1, t], 25], [[s, a0], 24], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, a1, s], [[[a0, t], 24], [[a1, t], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a2, s], [[[a2, t], 26], [[s, a0], 24], [[s, a1], 25], [[s, a3], 27]], 102], [[a0, a2, s], [[[a0, t], 24], [[a2, t], 26], [[s, a1], 25], [[s, a3], 27]], 102], [[a1, a2, s], [[[a1, t], 25], [[a2, t], 26], [[s, a0], 24], [[s, a3], 27]], 102], [[a0, a1, a2, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[s, a3], 27]], 102], [[a3, s], [[[a3, t], 27], [[s, a0], 24], [[s, a1], 25], [[s, a2], 26]], 102], [[a0, a3, s], [[[a0, t], 24], [[a3, t], 27], [[s, a1], 25], [[s, a2], 26]], 102], [[a1, a3, s], [[[a1, t], 25], [[a3, t], 27], [[s, a0], 24], [[s, a2], 26]], 102], [[a0, a1, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a3, t], 27], [[s, a2], 26]], 102], [[a2, a3, s], [[[a2, t], 26], [[a3, t], 27], [[s, a0], 24], [[s, a1], 25]], 102], [[a0, a2, a3, s], [[[a0, t], 24], [[a2, t], 26], [[a3, t], 27], [[s, a1], 25]], 102], [[a1, a2, a3, s], [[[a1, t], 25], [[a2, t], 26], [[a3, t], 27], [[s, a0], 24]], 102], [[a0, a1, a2, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[a3, t], 27]], 102]], 102, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, s], [[[a0, t], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a1, s], [[[a1, t], 25], [[s, a0], 24], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, a1, s], [[[a0, t], 24], [[a1, t], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a2, s], [[[a2, t], 26], [[s, a0], 24], [[s, a1], 25], [[s, a3], 27]], 102], [[a0, a2, s], [[[a0, t], 24], [[a2, t], 26], [[s, a1], 25], [[s, a3], 27]], 102], [[a1, a2, s], [[[a1, t], 25], [[a2, t], 26], [[s, a0], 24], [[s, a3], 27]], 102], [[a0, a1, a2, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[s, a3], 27]], 102], [[a3, s], [[[a3, t], 27], [[s, a0], 24], [[s, a1], 25], [[s, a2], 26]], 102], [[a0, a3, s], [[[a0, t], 24], [[a3, t], 27], [[s, a1], 25], [[s, a2], 26]], 102], [[a1, a3, s], [[[a1, t], 25], [[a3, t], 27], [[s, a0], 24], [[s, a2], 26]], 102], [[a0, a1, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a3, t], 27], [[s, a2], 26]], 102], [[a2, a3, s], [[[a2, t], 26], [[a3, t], 27], [[s, a0], 24], [[s, a1], 25]], 102], [[a0, a2, a3, s], [[[a0, t], 24], [[a2, t], 26], [[a3, t], 27], [[s, a1], 25]], 102], [[a1, a2, a3, s], [[[a1, t], 25], [[a2, t], 26], [[a3, t], 27], [[s, a0], 24]], 102], [[a0, a1, a2, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[a3, t], 27]], 102]], 102, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155073,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[24,24,25,25,26,26,27,27],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[24,1],[25,1],[26,1],[27,1],[24,1],[25,1],[26,1],[27,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0074","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[25,25,26,26,27,27,28,28,29,29]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, s], [[[a0, t], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, s], [[[a1, t], 26], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, s], [[[a0, t], 25], [[a1, t], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a2, s], [[[a2, t], 27], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a2, s], [[[a0, t], 25], [[a2, t], 27], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, a2, s], [[[a1, t], 26], [[a2, t], 27], [[s, a0], 25], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, a2, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a3, s], [[[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a3, s], [[[a0, t], 25], [[a3, t], 28], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a1, a3, s], [[[a1, t], 26], [[a3, t], 28], [[s, a0], 25], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a1, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[s, a2], 27], [[s, a4], 29]], 135], [[a2, a3, s], [[[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a4], 29]], 135], [[a0, a2, a3, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[s, a1], 26], [[s, a4], 29]], 135], [[a1, a2, a3, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a4], 29]], 135], [[a0, a1, a2, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a4], 29]], 135], [[a4, s], [[[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a4, s], [[[a0, t], 25], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a1, a4, s], [[[a1, t], 26], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a1, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a4, t], 29], [[s, a2], 27], [[s, a3], 28]], 135], [[a2, a4, s], [[[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28]], 135], [[a0, a2, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a4, t], 29], [[s, a1], 26], [[s, a3], 28]], 135], [[a1, a2, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a3], 28]], 135], [[a0, a1, a2, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a3], 28]], 135], [[a3, a4, s], [[[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27]], 135], [[a0, a3, a4, s], [[[a0, t], 25], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27]], 135], [[a1, a3, a4, s], [[[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27]], 135], [[a0, a1, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a2], 27]], 135], [[a2, a3, a4, s], [[[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26]], 135], [[a0, a2, a3, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26]], 135], [[a1, a2, a3, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25]], 135], [[a0, a1, a2, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29]], 135]], 135, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, s], [[[a0, t], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, s], [[[a1, t], 26], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, s], [[[a0, t], 25], [[a1, t], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a2, s], [[[a2, t], 27], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a2, s], [[[a0, t], 25], [[a2, t], 27], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, a2, s], [[[a1, t], 26], [[a2, t], 27], [[s, a0], 25], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, a2, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a3, s], [[[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a3, s], [[[a0, t], 25], [[a3, t], 28], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a1, a3, s], [[[a1, t], 26], [[a3, t], 28], [[s, a0], 25], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a1, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[s, a2], 27], [[s, a4], 29]], 135], [[a2, a3, s], [[[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a4], 29]], 135], [[a0, a2, a3, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[s, a1], 26], [[s, a4], 29]], 135], [[a1, a2, a3, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a4], 29]], 135], [[a0, a1, a2, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a4], 29]], 135], [[a4, s], [[[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a4, s], [[[a0, t], 25], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a1, a4, s], [[[a1, t], 26], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a1, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a4, t], 29], [[s, a2], 27], [[s, a3], 28]], 135], [[a2, a4, s], [[[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28]], 135], [[a0, a2, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a4, t], 29], [[s, a1], 26], [[s, a3], 28]], 135], [[a1, a2, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a3], 28]], 135], [[a0, a1, a2, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a3], 28]], 135], [[a3, a4, s], [[[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27]], 135], [[a0, a3, a4, s], [[[a0, t], 25], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27]], 135], [[a1, a3, a4, s], [[[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27]], 135], [[a0, a1, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a2], 27]], 135], [[a2, a3, a4, s], [[[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26]], 135], [[a0, a2, a3, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26]], 135], [[a1, a2, a3, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25]], 135], [[a0, a1, a2, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29]], 135]], 135, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155074,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[25,25,26,26,27,27,28,28,29,29],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[25,1],[26,1],[27,1],[28,1],[29,1],[25,1],[26,1],[27,1],[28,1],[29,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0075","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[26,26,27,27,31]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[s], [[[s, a0], 26], [[s, a1], 27]], 53], [[a0, s], [[[a0, t], 26], [[s, a1], 27]], 53], [[a1, s], [[[a1, t], 27], [[s, a0], 26]], 53], [[a0, a1, s], [[[a0, t], 26], [[a1, t], 27]], 53]], 53, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)","explanation_zh":"Wave6第九層核對=([[[s], [[[s, a0], 26], [[s, a1], 27]], 53], [[a0, s], [[[a0, t], 26], [[s, a1], 27]], 53], [[a1, s], [[[a1, t], 27], [[s, a0], 26]], 53], [[a0, a1, s], [[[a0, t], 26], [[a1, t], 27]], 53]], 53, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155075,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[26,26,27,27,31],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-minimum-cut-audit","semantic_template":"qf-mod46-mfmc-minimum-cut-audit/{\"capacities\":[[31,1],[26,1],[27,1],[26,1],[27,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0076","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[2,2,3,3,7],\"flows\":[2,2,3,3,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(5, 5, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155076,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[2,2,3,3,7],"flows":[2,2,3,3,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[7,1],[2,1],[3,1],[2,1],[3,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[0,1],[2,1],[3,1],[2,1],[3,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0077","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[3,3,4,4,5,5],\"flows\":[3,3,4,4,5,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 12, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(12, 12, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155077,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[3,3,4,4,5,5],"flows":[3,3,4,4,5,5],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[3,1],[4,1],[5,1],[3,1],[4,1],[5,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"flows\":[[3,1],[4,1],[5,1],[3,1],[4,1],[5,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0078","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[4,4,5,5,6,6,7,7],\"flows\":[4,4,5,5,6,6,7,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 22, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(22, 22, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155078,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[4,4,5,5,6,6,7,7],"flows":[4,4,5,5,6,6,7,7],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[4,1],[5,1],[6,1],[7,1],[4,1],[5,1],[6,1],[7,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"flows\":[[4,1],[5,1],[6,1],[7,1],[4,1],[5,1],[6,1],[7,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0079","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[5,5,6,6,7,7,8,8,9,9,10],\"flows\":[5,5,6,6,7,7,8,8,9,9,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(35, 35, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(35, 35, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155079,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"],["a0","s"]],"source":"s","sink":"t","capacities":[5,5,6,6,7,7,8,8,9,9,10],"flows":[5,5,6,6,7,7,8,8,9,9,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[10,1],[5,1],[6,1],[7,1],[8,1],[9,1],[5,1],[6,1],[7,1],[8,1],[9,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"flows\":[[0,1],[5,1],[6,1],[7,1],[8,1],[9,1],[5,1],[6,1],[7,1],[8,1],[9,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0080","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[6,6,7,7],\"flows\":[6,6,7,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 13, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(13, 13, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155080,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[6,6,7,7],"flows":[6,6,7,7],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[6,1],[7,1],[6,1],[7,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[6,1],[7,1],[6,1],[7,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0081","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[7,7,8,8,9,9],\"flows\":[7,7,8,8,9,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 24, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(24, 24, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155081,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[7,7,8,8,9,9],"flows":[7,7,8,8,9,9],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[7,1],[8,1],[9,1],[7,1],[8,1],[9,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"flows\":[[7,1],[8,1],[9,1],[7,1],[8,1],[9,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0082","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[8,8,9,9,10,10,11,11,13],\"flows\":[8,8,9,9,10,10,11,11,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(38, 38, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(38, 38, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155082,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[8,8,9,9,10,10,11,11,13],"flows":[8,8,9,9,10,10,11,11,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[13,1],[8,1],[9,1],[10,1],[11,1],[8,1],[9,1],[10,1],[11,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"flows\":[[0,1],[8,1],[9,1],[10,1],[11,1],[8,1],[9,1],[10,1],[11,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0083","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[9,9,10,10,11,11,12,12,13,13],\"flows\":[9,9,10,10,11,11,12,12,13,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(55, 55, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(55, 55, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155083,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[9,9,10,10,11,11,12,12,13,13],"flows":[9,9,10,10,11,11,12,12,13,13],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[9,1],[10,1],[11,1],[12,1],[13,1],[9,1],[10,1],[11,1],[12,1],[13,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"flows\":[[9,1],[10,1],[11,1],[12,1],[13,1],[9,1],[10,1],[11,1],[12,1],[13,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0084","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[10,10,11,11],\"flows\":[10,10,11,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 21, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(21, 21, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155084,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[10,10,11,11],"flows":[10,10,11,11],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[10,1],[11,1],[10,1],[11,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[10,1],[11,1],[10,1],[11,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0085","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[11,11,12,12,13,13,16],\"flows\":[11,11,12,12,13,13,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(36, 36, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(36, 36, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155085,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[11,11,12,12,13,13,16],"flows":[11,11,12,12,13,13,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[16,1],[11,1],[12,1],[13,1],[11,1],[12,1],[13,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"flows\":[[0,1],[11,1],[12,1],[13,1],[11,1],[12,1],[13,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0086","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[12,12,13,13,14,14,15,15],\"flows\":[12,12,13,13,14,14,15,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(54, 54, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(54, 54, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155086,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[12,12,13,13,14,14,15,15],"flows":[12,12,13,13,14,14,15,15],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[12,1],[13,1],[14,1],[15,1],[12,1],[13,1],[14,1],[15,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"flows\":[[12,1],[13,1],[14,1],[15,1],[12,1],[13,1],[14,1],[15,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0087","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[13,13,14,14,15,15,16,16,17,17],\"flows\":[13,13,14,14,15,15,16,16,17,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(75, 75, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(75, 75, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155087,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[13,13,14,14,15,15,16,16,17,17],"flows":[13,13,14,14,15,15,16,16,17,17],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[13,1],[14,1],[15,1],[16,1],[17,1],[13,1],[14,1],[15,1],[16,1],[17,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"flows\":[[13,1],[14,1],[15,1],[16,1],[17,1],[13,1],[14,1],[15,1],[16,1],[17,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0088","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[14,14,15,15,19],\"flows\":[14,14,15,15,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 29, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(29, 29, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155088,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[14,14,15,15,19],"flows":[14,14,15,15,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[19,1],[14,1],[15,1],[14,1],[15,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[0,1],[14,1],[15,1],[14,1],[15,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0089","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[15,15,16,16,17,17],\"flows\":[15,15,16,16,17,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(48, 48, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(48, 48, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155089,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[15,15,16,16,17,17],"flows":[15,15,16,16,17,17],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[15,1],[16,1],[17,1],[15,1],[16,1],[17,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"flows\":[[15,1],[16,1],[17,1],[15,1],[16,1],[17,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0090","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[16,16,17,17,18,18,19,19],\"flows\":[16,16,17,17,18,18,19,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(70, 70, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(70, 70, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155090,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[16,16,17,17,18,18,19,19],"flows":[16,16,17,17,18,18,19,19],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[16,1],[17,1],[18,1],[19,1],[16,1],[17,1],[18,1],[19,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"flows\":[[16,1],[17,1],[18,1],[19,1],[16,1],[17,1],[18,1],[19,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0091","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[17,17,18,18,19,19,20,20,21,21,22],\"flows\":[17,17,18,18,19,19,20,20,21,21,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(95, 95, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(95, 95, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155091,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"],["a0","s"]],"source":"s","sink":"t","capacities":[17,17,18,18,19,19,20,20,21,21,22],"flows":[17,17,18,18,19,19,20,20,21,21,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[22,1],[17,1],[18,1],[19,1],[20,1],[21,1],[17,1],[18,1],[19,1],[20,1],[21,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"flows\":[[0,1],[17,1],[18,1],[19,1],[20,1],[21,1],[17,1],[18,1],[19,1],[20,1],[21,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0092","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[18,18,19,19],\"flows\":[18,18,19,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(37, 37, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(37, 37, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155092,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[18,18,19,19],"flows":[18,18,19,19],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[18,1],[19,1],[18,1],[19,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[18,1],[19,1],[18,1],[19,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0093","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[19,19,20,20,21,21],\"flows\":[19,19,20,20,21,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(60, 60, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(60, 60, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155093,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[19,19,20,20,21,21],"flows":[19,19,20,20,21,21],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[19,1],[20,1],[21,1],[19,1],[20,1],[21,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"flows\":[[19,1],[20,1],[21,1],[19,1],[20,1],[21,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0094","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[20,20,21,21,22,22,23,23,25],\"flows\":[20,20,21,21,22,22,23,23,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(86, 86, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(86, 86, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155094,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[20,20,21,21,22,22,23,23,25],"flows":[20,20,21,21,22,22,23,23,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[25,1],[20,1],[21,1],[22,1],[23,1],[20,1],[21,1],[22,1],[23,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"flows\":[[0,1],[20,1],[21,1],[22,1],[23,1],[20,1],[21,1],[22,1],[23,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0095","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[21,21,22,22,23,23,24,24,25,25],\"flows\":[21,21,22,22,23,23,24,24,25,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(115, 115, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(115, 115, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155095,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[21,21,22,22,23,23,24,24,25,25],"flows":[21,21,22,22,23,23,24,24,25,25],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[21,1],[22,1],[23,1],[24,1],[25,1],[21,1],[22,1],[23,1],[24,1],[25,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"flows\":[[21,1],[22,1],[23,1],[24,1],[25,1],[21,1],[22,1],[23,1],[24,1],[25,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0096","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[22,22,23,23],\"flows\":[22,22,23,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(45, 45, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(45, 45, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155096,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[22,22,23,23],"flows":[22,22,23,23],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[22,1],[23,1],[22,1],[23,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[22,1],[23,1],[22,1],[23,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0097","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[23,23,24,24,25,25,28],\"flows\":[23,23,24,24,25,25,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(72, 72, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(72, 72, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155097,"parameters":{"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[23,23,24,24,25,25,28],"flows":[23,23,24,24,25,25,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[28,1],[23,1],[24,1],[25,1],[23,1],[24,1],[25,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"]],\"flows\":[[0,1],[23,1],[24,1],[25,1],[23,1],[24,1],[25,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0098","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[24,24,25,25,26,26,27,27],\"flows\":[24,24,25,25,26,26,27,27]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(102, 102, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(102, 102, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155098,"parameters":{"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[24,24,25,25,26,26,27,27],"flows":[24,24,25,25,26,26,27,27],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[24,1],[25,1],[26,1],[27,1],[24,1],[25,1],[26,1],[27,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"]],\"flows\":[[24,1],[25,1],[26,1],[27,1],[24,1],[25,1],[26,1],[27,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0099","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"s\",\"a2\"],[\"a2\",\"t\"],[\"s\",\"a3\"],[\"a3\",\"t\"],[\"s\",\"a4\"],[\"a4\",\"t\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[25,25,26,26,27,27,28,28,29,29],\"flows\":[25,25,26,26,27,27,28,28,29,29]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(135, 135, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(135, 135, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155099,"parameters":{"vertices":["s","a0","a1","a2","a3","a4","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["s","a4"],["a4","t"]],"source":"s","sink":"t","capacities":[25,25,26,26,27,27,28,28,29,29],"flows":[25,25,26,26,27,27,28,28,29,29],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[25,1],[26,1],[27,1],[28,1],[29,1],[25,1],[26,1],[27,1],[28,1],[29,1]],\"edges\":[[\"a0\",\"t\"],[\"a1\",\"t\"],[\"a2\",\"t\"],[\"a3\",\"t\"],[\"a4\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"],[\"s\",\"a2\"],[\"s\",\"a3\"],[\"s\",\"a4\"]],\"flows\":[[25,1],[26,1],[27,1],[28,1],[29,1],[25,1],[26,1],[27,1],[28,1],[29,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"a2\",\"a3\",\"a4\",\"s\",\"t\"]}"}},"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-r05-max-flow-min-cut-theorem-0100","batch_id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-max-flow-min-cut-theorem"],"title_zh":"最大流最小割定理","stem_zh":"完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={\"vertices\":[\"s\",\"a0\",\"a1\",\"t\"],\"edges\":[[\"s\",\"a0\"],[\"a0\",\"t\"],[\"s\",\"a1\"],[\"a1\",\"t\"],[\"a0\",\"s\"]],\"source\":\"s\",\"sink\":\"t\",\"capacities\":[26,26,27,27,31],\"flows\":[26,26,27,27,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(53, 53, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation_zh":"Wave6第九層核對=(53, 53, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["最大流最小割定理"]},"variant":{"family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155100,"parameters":{"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[26,26,27,27,31],"flows":[26,26,27,27,0],"domain":"Complete_finite_max_flow_min_cut","operation":"mfmc-equal-value-optimality","semantic_template":"qf-mod46-mfmc-equal-value-optimality/{\"capacities\":[[31,1],[26,1],[27,1],[26,1],[27,1]],\"edges\":[[\"a0\",\"s\"],[\"a0\",\"t\"],[\"a1\",\"t\"],[\"s\",\"a0\"],[\"s\",\"a1\"]],\"flows\":[[0,1],[26,1],[27,1],[26,1],[27,1]],\"sink\":\"t\",\"source\":\"s\",\"vertices\":[\"a0\",\"a1\",\"s\",\"t\"]}"}},"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"}}