{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ae737099023056dcc70511865395c73785fbc7479548ada4f24035d1147a31f","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155001,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[2, 2, 3, 3, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6a23b7159d3d61efd870132fad82691f8932070e84d5d414bec1969e61fe107","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155002,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[3, 3, 4, 4, 5, 5]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb31e3e73d4b311b72e8a61ad7de5949a0196c8bfde739e57a40cc44132c8cbe","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155003,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[4, 4, 5, 5, 6, 6, 7, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d902a80d3bcb10df5c8acfc32e4b2d1af808dab4b606db71e0399d7399ae8d9b","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155004,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebb0ed6a704d87923aa285801612451b38209c9726f670ba30a37fdaa59c8ab6","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155005,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[6, 6, 7, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67349e5242d91e01d41975b7b726f71fd55218bc0943933416f6b30bc259d644","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155006,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[7, 7, 8, 8, 9, 9]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad65433d52b6f0b19a6a13a0449adb80e8f4b3edb23e3fdaca911c683f43c527","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155007,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[8, 8, 9, 9, 10, 10, 11, 11, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4de1a0766af89515f3e3139cc4bb286b93c3bb78e17ccbbd974f613807abd4da","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155008,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"244192531a1ad0fe578ea0c308951a083f9989c617098799d4f4b45fbb458fa3","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155009,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[10, 10, 11, 11]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dd78867aa499582afc93d58aac991c8e4b79feb2fedc7a16e03a02955de93c7","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155010,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[11, 11, 12, 12, 13, 13, 16]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"889aefc85c3bcfb5d193a070c514989a437ad5d444edc1b64c34fc67cb62c51f","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155011,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[12, 12, 13, 13, 14, 14, 15, 15]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9214827cf656ef74dc11eae25aa09258737f1c48b26e12cb98e1f7ee434cd234","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155012,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"112bc19f4fa4fc07fd314ed1e88462bd70a84649cb34571c0eeb45429c35fd4a","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155013,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[14, 14, 15, 15, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0f325c02ecef3dc4a709409d638632cb4db027224efa0028c5d4064890d8d12","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155014,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fb44d11477b86a499d0d3ab3db6ac34d8854a5e10ff25b2ebbe4eeca097e368","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155015,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[16, 16, 17, 17, 18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3bbd32977d2e38ba96c97899e9dea1b233ac5fb2e0d8cd1772fae80076a2c9b","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155016,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24954c3321c18156fbcfdce30f7250c4a614cab4b01d23aa1abce2236ca02bcc","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155017,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09c834a272abff1515c461035c28eba91cf59467944ddf27d28a07655eab7c3a","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155018,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[19, 19, 20, 20, 21, 21]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2620f26f9113a09cf1453f1c7e88dca47707e7fb26a0eb23f39cf4e099f08b5d","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155019,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[20, 20, 21, 21, 22, 22, 23, 23, 25]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f47623cfc35cfaf18def69798bfe48e898b43ed5236244861a61b16dca91668","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155020,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"752e7f71cdde4a6bf58c4eeecdfa9a5a49dc2b02669f612a06eea07bc43b6be7","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155021,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[22, 22, 23, 23]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26785fa150b65b8d869cbb7e8f84d27acb396dcbc95ae077df18e54beaf1e150","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155022,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[23, 23, 24, 24, 25, 25, 28]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f50165cd6947f705db8c7e03289274e56a1464478c5cfb4ee7fe0c3d229624c","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155023,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[24, 24, 25, 25, 26, 26, 27, 27]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8a69aaee580725dcab2089c7a0979772d55c75325e7935338ddf31d88c4c89a","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155024,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"926c25d0f8be835d1614129a1a473ae23d2f6a267ed4fa3b7acc3fa871dd3903","family_id":"qf-mod46-mfmc-complete-cut-enumeration","seed":202611155025,"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\"]}"},"fields":{"title":"Every Source-Side Vertex Subset Defines One Directed Cut Audit","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[26, 26, 27, 27, 31]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).","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":"Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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)."},"skills":["Every Source-Side Vertex Subset Defines One Directed Cut Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01f7ea00a9f7b170de86d39ca2eee0f453ae78f8ed8e17855c3ef4bf9affec08","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155026,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[2, 2, 3, 3, 7], candidate flows=[1, 1, 2, 2, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(3, 3, [[[s], 3, 0, 3, 5, Yes, Yes], [[a0, s], 3, 0, 3, 5, Yes, Yes], [[a1, s], 3, 0, 3, 5, Yes, Yes], [[a0, a1, s], 3, 0, 3, 5, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (3, 3, [[[s], 3, 0, 3, 5, Yes, Yes], [[a0, s], 3, 0, 3, 5, Yes, Yes], [[a1, s], 3, 0, 3, 5, Yes, Yes], [[a0, a1, s], 3, 0, 3, 5, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a129c06a2b2a8992b7bde45a121e6a4ea5e6fcebba4048dfb5183f150c155217","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155027,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[3, 3, 4, 4, 5, 5], candidate flows=[2, 2, 3, 3, 4, 4]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(9, 9, [[[s], 9, 0, 9, 12, Yes, Yes], [[a0, s], 9, 0, 9, 12, Yes, Yes], [[a1, s], 9, 0, 9, 12, Yes, Yes], [[a0, a1, s], 9, 0, 9, 12, Yes, Yes], [[a2, s], 9, 0, 9, 12, Yes, Yes], [[a0, a2, s], 9, 0, 9, 12, Yes, Yes], [[a1, a2, s], 9, 0, 9, 12, Yes, Yes], [[a0, a1, a2, s], 9, 0, 9, 12, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (9, 9, [[[s], 9, 0, 9, 12, Yes, Yes], [[a0, s], 9, 0, 9, 12, Yes, Yes], [[a1, s], 9, 0, 9, 12, Yes, Yes], [[a0, a1, s], 9, 0, 9, 12, Yes, Yes], [[a2, s], 9, 0, 9, 12, Yes, Yes], [[a0, a2, s], 9, 0, 9, 12, Yes, Yes], [[a1, a2, s], 9, 0, 9, 12, Yes, Yes], [[a0, a1, a2, s], 9, 0, 9, 12, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e377809f097f9a51a05159b45ad2c5d1fddf239d87df3ada34698273f21431b","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155028,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[4, 4, 5, 5, 6, 6, 7, 7], candidate flows=[3, 3, 4, 4, 5, 5, 6, 6]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(18, 18, [[[s], 18, 0, 18, 22, Yes, Yes], [[a0, s], 18, 0, 18, 22, Yes, Yes], [[a1, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, s], 18, 0, 18, 22, Yes, Yes], [[a2, s], 18, 0, 18, 22, Yes, Yes], [[a0, a2, s], 18, 0, 18, 22, Yes, Yes], [[a1, a2, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a2, s], 18, 0, 18, 22, Yes, Yes], [[a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a3, s], 18, 0, 18, 22, Yes, Yes], [[a1, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a3, s], 18, 0, 18, 22, Yes, Yes], [[a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a1, a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a2, a3, s], 18, 0, 18, 22, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (18, 18, [[[s], 18, 0, 18, 22, Yes, Yes], [[a0, s], 18, 0, 18, 22, Yes, Yes], [[a1, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, s], 18, 0, 18, 22, Yes, Yes], [[a2, s], 18, 0, 18, 22, Yes, Yes], [[a0, a2, s], 18, 0, 18, 22, Yes, Yes], [[a1, a2, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a2, s], 18, 0, 18, 22, Yes, Yes], [[a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a3, s], 18, 0, 18, 22, Yes, Yes], [[a1, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a3, s], 18, 0, 18, 22, Yes, Yes], [[a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a1, a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a2, a3, s], 18, 0, 18, 22, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e0cecb4a18a74b5b891542c157ee2efc0372e539bb19c419e3bd3ba28f7657b","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155029,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10], candidate flows=[4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(30, 30, [[[s], 30, 0, 30, 35, Yes, Yes], [[a0, s], 30, 0, 30, 35, Yes, Yes], [[a1, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, s], 30, 0, 30, 35, Yes, Yes], [[a2, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, s], 30, 0, 30, 35, Yes, Yes], [[a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a3, s], 30, 0, 30, 35, Yes, Yes], [[a1, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a3, s], 30, 0, 30, 35, Yes, Yes], [[a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a4, s], 30, 0, 30, 35, Yes, Yes], [[a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (30, 30, [[[s], 30, 0, 30, 35, Yes, Yes], [[a0, s], 30, 0, 30, 35, Yes, Yes], [[a1, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, s], 30, 0, 30, 35, Yes, Yes], [[a2, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, s], 30, 0, 30, 35, Yes, Yes], [[a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a3, s], 30, 0, 30, 35, Yes, Yes], [[a1, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a3, s], 30, 0, 30, 35, Yes, Yes], [[a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a4, s], 30, 0, 30, 35, Yes, Yes], [[a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e5e541792e75013ebefdbdc524fef57cba1ef2a80bcfab6f022b3ef80097133","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155030,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[6, 6, 7, 7], candidate flows=[5, 5, 6, 6]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(11, 11, [[[s], 11, 0, 11, 13, Yes, Yes], [[a0, s], 11, 0, 11, 13, Yes, Yes], [[a1, s], 11, 0, 11, 13, Yes, Yes], [[a0, a1, s], 11, 0, 11, 13, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (11, 11, [[[s], 11, 0, 11, 13, Yes, Yes], [[a0, s], 11, 0, 11, 13, Yes, Yes], [[a1, s], 11, 0, 11, 13, Yes, Yes], [[a0, a1, s], 11, 0, 11, 13, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60937caa0da243dadf6bb39b8bd1ef730f75cf456e8e40cc57ec1cf33cf07149","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155031,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[7, 7, 8, 8, 9, 9], candidate flows=[6, 6, 7, 7, 8, 8]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(21, 21, [[[s], 21, 0, 21, 24, Yes, Yes], [[a0, s], 21, 0, 21, 24, Yes, Yes], [[a1, s], 21, 0, 21, 24, Yes, Yes], [[a0, a1, s], 21, 0, 21, 24, Yes, Yes], [[a2, s], 21, 0, 21, 24, Yes, Yes], [[a0, a2, s], 21, 0, 21, 24, Yes, Yes], [[a1, a2, s], 21, 0, 21, 24, Yes, Yes], [[a0, a1, a2, s], 21, 0, 21, 24, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (21, 21, [[[s], 21, 0, 21, 24, Yes, Yes], [[a0, s], 21, 0, 21, 24, Yes, Yes], [[a1, s], 21, 0, 21, 24, Yes, Yes], [[a0, a1, s], 21, 0, 21, 24, Yes, Yes], [[a2, s], 21, 0, 21, 24, Yes, Yes], [[a0, a2, s], 21, 0, 21, 24, Yes, Yes], [[a1, a2, s], 21, 0, 21, 24, Yes, Yes], [[a0, a1, a2, s], 21, 0, 21, 24, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06cad729aa9f4e12c5f94252322ff2cbe6c36021fc083175b499bc96d962d82f","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155032,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[8, 8, 9, 9, 10, 10, 11, 11, 13], candidate flows=[7, 7, 8, 8, 9, 9, 10, 10, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(34, 34, [[[s], 34, 0, 34, 38, Yes, Yes], [[a0, s], 34, 0, 34, 38, Yes, Yes], [[a1, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, s], 34, 0, 34, 38, Yes, Yes], [[a2, s], 34, 0, 34, 38, Yes, Yes], [[a0, a2, s], 34, 0, 34, 38, Yes, Yes], [[a1, a2, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a2, s], 34, 0, 34, 38, Yes, Yes], [[a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a3, s], 34, 0, 34, 38, Yes, Yes], [[a1, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a3, s], 34, 0, 34, 38, Yes, Yes], [[a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a1, a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a2, a3, s], 34, 0, 34, 38, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (34, 34, [[[s], 34, 0, 34, 38, Yes, Yes], [[a0, s], 34, 0, 34, 38, Yes, Yes], [[a1, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, s], 34, 0, 34, 38, Yes, Yes], [[a2, s], 34, 0, 34, 38, Yes, Yes], [[a0, a2, s], 34, 0, 34, 38, Yes, Yes], [[a1, a2, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a2, s], 34, 0, 34, 38, Yes, Yes], [[a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a3, s], 34, 0, 34, 38, Yes, Yes], [[a1, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a3, s], 34, 0, 34, 38, Yes, Yes], [[a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a1, a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a2, a3, s], 34, 0, 34, 38, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83dbee6701e5c401c7863ce65e55dcb4716aaa22bc1abffe1669310249d6a31f","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155033,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13], candidate flows=[8, 8, 9, 9, 10, 10, 11, 11, 12, 12]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(50, 50, [[[s], 50, 0, 50, 55, Yes, Yes], [[a0, s], 50, 0, 50, 55, Yes, Yes], [[a1, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, s], 50, 0, 50, 55, Yes, Yes], [[a2, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, s], 50, 0, 50, 55, Yes, Yes], [[a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a3, s], 50, 0, 50, 55, Yes, Yes], [[a1, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a3, s], 50, 0, 50, 55, Yes, Yes], [[a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a4, s], 50, 0, 50, 55, Yes, Yes], [[a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (50, 50, [[[s], 50, 0, 50, 55, Yes, Yes], [[a0, s], 50, 0, 50, 55, Yes, Yes], [[a1, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, s], 50, 0, 50, 55, Yes, Yes], [[a2, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, s], 50, 0, 50, 55, Yes, Yes], [[a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a3, s], 50, 0, 50, 55, Yes, Yes], [[a1, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a3, s], 50, 0, 50, 55, Yes, Yes], [[a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a4, s], 50, 0, 50, 55, Yes, Yes], [[a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"583124badf7c6918d833ff8d8856581664bd15f018294de1375f8243e0010222","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155034,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[10, 10, 11, 11], candidate flows=[9, 9, 10, 10]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(19, 19, [[[s], 19, 0, 19, 21, Yes, Yes], [[a0, s], 19, 0, 19, 21, Yes, Yes], [[a1, s], 19, 0, 19, 21, Yes, Yes], [[a0, a1, s], 19, 0, 19, 21, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (19, 19, [[[s], 19, 0, 19, 21, Yes, Yes], [[a0, s], 19, 0, 19, 21, Yes, Yes], [[a1, s], 19, 0, 19, 21, Yes, Yes], [[a0, a1, s], 19, 0, 19, 21, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23d03a37cd72de0ff2f6b5a7527f7badd6b1109d172f18070bb18d92aa2f217e","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155035,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[11, 11, 12, 12, 13, 13, 16], candidate flows=[10, 10, 11, 11, 12, 12, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(33, 33, [[[s], 33, 0, 33, 36, Yes, Yes], [[a0, s], 33, 0, 33, 36, Yes, Yes], [[a1, s], 33, 0, 33, 36, Yes, Yes], [[a0, a1, s], 33, 0, 33, 36, Yes, Yes], [[a2, s], 33, 0, 33, 36, Yes, Yes], [[a0, a2, s], 33, 0, 33, 36, Yes, Yes], [[a1, a2, s], 33, 0, 33, 36, Yes, Yes], [[a0, a1, a2, s], 33, 0, 33, 36, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (33, 33, [[[s], 33, 0, 33, 36, Yes, Yes], [[a0, s], 33, 0, 33, 36, Yes, Yes], [[a1, s], 33, 0, 33, 36, Yes, Yes], [[a0, a1, s], 33, 0, 33, 36, Yes, Yes], [[a2, s], 33, 0, 33, 36, Yes, Yes], [[a0, a2, s], 33, 0, 33, 36, Yes, Yes], [[a1, a2, s], 33, 0, 33, 36, Yes, Yes], [[a0, a1, a2, s], 33, 0, 33, 36, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e96c39d085ed9561e59b4665a2eedda2e470a283f8c6f9756b4f001883fe728","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155036,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[12, 12, 13, 13, 14, 14, 15, 15], candidate flows=[11, 11, 12, 12, 13, 13, 14, 14]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(50, 50, [[[s], 50, 0, 50, 54, Yes, Yes], [[a0, s], 50, 0, 50, 54, Yes, Yes], [[a1, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, s], 50, 0, 50, 54, Yes, Yes], [[a2, s], 50, 0, 50, 54, Yes, Yes], [[a0, a2, s], 50, 0, 50, 54, Yes, Yes], [[a1, a2, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a2, s], 50, 0, 50, 54, Yes, Yes], [[a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a3, s], 50, 0, 50, 54, Yes, Yes], [[a1, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a3, s], 50, 0, 50, 54, Yes, Yes], [[a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a1, a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a2, a3, s], 50, 0, 50, 54, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (50, 50, [[[s], 50, 0, 50, 54, Yes, Yes], [[a0, s], 50, 0, 50, 54, Yes, Yes], [[a1, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, s], 50, 0, 50, 54, Yes, Yes], [[a2, s], 50, 0, 50, 54, Yes, Yes], [[a0, a2, s], 50, 0, 50, 54, Yes, Yes], [[a1, a2, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a2, s], 50, 0, 50, 54, Yes, Yes], [[a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a3, s], 50, 0, 50, 54, Yes, Yes], [[a1, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a3, s], 50, 0, 50, 54, Yes, Yes], [[a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a1, a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a2, a3, s], 50, 0, 50, 54, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29702653f1522be8cb763692924185ec57f5129e52052f753e21e4e07ee243a5","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155037,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17], candidate flows=[12, 12, 13, 13, 14, 14, 15, 15, 16, 16]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(70, 70, [[[s], 70, 0, 70, 75, Yes, Yes], [[a0, s], 70, 0, 70, 75, Yes, Yes], [[a1, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, s], 70, 0, 70, 75, Yes, Yes], [[a2, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, s], 70, 0, 70, 75, Yes, Yes], [[a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a3, s], 70, 0, 70, 75, Yes, Yes], [[a1, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a3, s], 70, 0, 70, 75, Yes, Yes], [[a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a4, s], 70, 0, 70, 75, Yes, Yes], [[a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (70, 70, [[[s], 70, 0, 70, 75, Yes, Yes], [[a0, s], 70, 0, 70, 75, Yes, Yes], [[a1, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, s], 70, 0, 70, 75, Yes, Yes], [[a2, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, s], 70, 0, 70, 75, Yes, Yes], [[a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a3, s], 70, 0, 70, 75, Yes, Yes], [[a1, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a3, s], 70, 0, 70, 75, Yes, Yes], [[a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a4, s], 70, 0, 70, 75, Yes, Yes], [[a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43527d34394e847dbbe1520ab64e0caac876fe5c2a2e8f316263b03348164a69","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155038,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[14, 14, 15, 15, 19], candidate flows=[13, 13, 14, 14, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(27, 27, [[[s], 27, 0, 27, 29, Yes, Yes], [[a0, s], 27, 0, 27, 29, Yes, Yes], [[a1, s], 27, 0, 27, 29, Yes, Yes], [[a0, a1, s], 27, 0, 27, 29, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (27, 27, [[[s], 27, 0, 27, 29, Yes, Yes], [[a0, s], 27, 0, 27, 29, Yes, Yes], [[a1, s], 27, 0, 27, 29, Yes, Yes], [[a0, a1, s], 27, 0, 27, 29, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ee9c543eb4b817a3651b5f5e540161760e17132ee2667060fb911e53c213289","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155039,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[15, 15, 16, 16, 17, 17], candidate flows=[14, 14, 15, 15, 16, 16]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(45, 45, [[[s], 45, 0, 45, 48, Yes, Yes], [[a0, s], 45, 0, 45, 48, Yes, Yes], [[a1, s], 45, 0, 45, 48, Yes, Yes], [[a0, a1, s], 45, 0, 45, 48, Yes, Yes], [[a2, s], 45, 0, 45, 48, Yes, Yes], [[a0, a2, s], 45, 0, 45, 48, Yes, Yes], [[a1, a2, s], 45, 0, 45, 48, Yes, Yes], [[a0, a1, a2, s], 45, 0, 45, 48, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (45, 45, [[[s], 45, 0, 45, 48, Yes, Yes], [[a0, s], 45, 0, 45, 48, Yes, Yes], [[a1, s], 45, 0, 45, 48, Yes, Yes], [[a0, a1, s], 45, 0, 45, 48, Yes, Yes], [[a2, s], 45, 0, 45, 48, Yes, Yes], [[a0, a2, s], 45, 0, 45, 48, Yes, Yes], [[a1, a2, s], 45, 0, 45, 48, Yes, Yes], [[a0, a1, a2, s], 45, 0, 45, 48, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e80d617731e0c641cf014d3b13e623f7c02b1d7ab15267317e065973ddc5dbfa","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155040,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[16, 16, 17, 17, 18, 18, 19, 19], candidate flows=[15, 15, 16, 16, 17, 17, 18, 18]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(66, 66, [[[s], 66, 0, 66, 70, Yes, Yes], [[a0, s], 66, 0, 66, 70, Yes, Yes], [[a1, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, s], 66, 0, 66, 70, Yes, Yes], [[a2, s], 66, 0, 66, 70, Yes, Yes], [[a0, a2, s], 66, 0, 66, 70, Yes, Yes], [[a1, a2, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a2, s], 66, 0, 66, 70, Yes, Yes], [[a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a3, s], 66, 0, 66, 70, Yes, Yes], [[a1, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a3, s], 66, 0, 66, 70, Yes, Yes], [[a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a1, a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a2, a3, s], 66, 0, 66, 70, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (66, 66, [[[s], 66, 0, 66, 70, Yes, Yes], [[a0, s], 66, 0, 66, 70, Yes, Yes], [[a1, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, s], 66, 0, 66, 70, Yes, Yes], [[a2, s], 66, 0, 66, 70, Yes, Yes], [[a0, a2, s], 66, 0, 66, 70, Yes, Yes], [[a1, a2, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a2, s], 66, 0, 66, 70, Yes, Yes], [[a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a3, s], 66, 0, 66, 70, Yes, Yes], [[a1, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a3, s], 66, 0, 66, 70, Yes, Yes], [[a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a1, a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a2, a3, s], 66, 0, 66, 70, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d902905f5810ae8fc6e7c09f9a5ac49f7d1194c386b9d59ab80d3e6ec8766f0","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155041,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22], candidate flows=[16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(90, 90, [[[s], 90, 0, 90, 95, Yes, Yes], [[a0, s], 90, 0, 90, 95, Yes, Yes], [[a1, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, s], 90, 0, 90, 95, Yes, Yes], [[a2, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, s], 90, 0, 90, 95, Yes, Yes], [[a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a3, s], 90, 0, 90, 95, Yes, Yes], [[a1, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a3, s], 90, 0, 90, 95, Yes, Yes], [[a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a4, s], 90, 0, 90, 95, Yes, Yes], [[a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (90, 90, [[[s], 90, 0, 90, 95, Yes, Yes], [[a0, s], 90, 0, 90, 95, Yes, Yes], [[a1, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, s], 90, 0, 90, 95, Yes, Yes], [[a2, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, s], 90, 0, 90, 95, Yes, Yes], [[a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a3, s], 90, 0, 90, 95, Yes, Yes], [[a1, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a3, s], 90, 0, 90, 95, Yes, Yes], [[a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a4, s], 90, 0, 90, 95, Yes, Yes], [[a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a73d5f0cd827c2c13575684a8e94691456ed757ead30d95f79a8f0c98f807b0c","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155042,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[18, 18, 19, 19], candidate flows=[17, 17, 18, 18]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(35, 35, [[[s], 35, 0, 35, 37, Yes, Yes], [[a0, s], 35, 0, 35, 37, Yes, Yes], [[a1, s], 35, 0, 35, 37, Yes, Yes], [[a0, a1, s], 35, 0, 35, 37, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (35, 35, [[[s], 35, 0, 35, 37, Yes, Yes], [[a0, s], 35, 0, 35, 37, Yes, Yes], [[a1, s], 35, 0, 35, 37, Yes, Yes], [[a0, a1, s], 35, 0, 35, 37, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c991adc880280a3846b28c681dea8305891b15e5ef4782d53eca78a7e465dc44","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155043,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[19, 19, 20, 20, 21, 21], candidate flows=[18, 18, 19, 19, 20, 20]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(57, 57, [[[s], 57, 0, 57, 60, Yes, Yes], [[a0, s], 57, 0, 57, 60, Yes, Yes], [[a1, s], 57, 0, 57, 60, Yes, Yes], [[a0, a1, s], 57, 0, 57, 60, Yes, Yes], [[a2, s], 57, 0, 57, 60, Yes, Yes], [[a0, a2, s], 57, 0, 57, 60, Yes, Yes], [[a1, a2, s], 57, 0, 57, 60, Yes, Yes], [[a0, a1, a2, s], 57, 0, 57, 60, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (57, 57, [[[s], 57, 0, 57, 60, Yes, Yes], [[a0, s], 57, 0, 57, 60, Yes, Yes], [[a1, s], 57, 0, 57, 60, Yes, Yes], [[a0, a1, s], 57, 0, 57, 60, Yes, Yes], [[a2, s], 57, 0, 57, 60, Yes, Yes], [[a0, a2, s], 57, 0, 57, 60, Yes, Yes], [[a1, a2, s], 57, 0, 57, 60, Yes, Yes], [[a0, a1, a2, s], 57, 0, 57, 60, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c47a06bc2594bd7c1115e891e54b20378b8ba0da88cded7c023b87cde0098b3","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155044,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[20, 20, 21, 21, 22, 22, 23, 23, 25], candidate flows=[19, 19, 20, 20, 21, 21, 22, 22, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(82, 82, [[[s], 82, 0, 82, 86, Yes, Yes], [[a0, s], 82, 0, 82, 86, Yes, Yes], [[a1, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, s], 82, 0, 82, 86, Yes, Yes], [[a2, s], 82, 0, 82, 86, Yes, Yes], [[a0, a2, s], 82, 0, 82, 86, Yes, Yes], [[a1, a2, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a2, s], 82, 0, 82, 86, Yes, Yes], [[a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a3, s], 82, 0, 82, 86, Yes, Yes], [[a1, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a3, s], 82, 0, 82, 86, Yes, Yes], [[a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a1, a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a2, a3, s], 82, 0, 82, 86, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (82, 82, [[[s], 82, 0, 82, 86, Yes, Yes], [[a0, s], 82, 0, 82, 86, Yes, Yes], [[a1, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, s], 82, 0, 82, 86, Yes, Yes], [[a2, s], 82, 0, 82, 86, Yes, Yes], [[a0, a2, s], 82, 0, 82, 86, Yes, Yes], [[a1, a2, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a2, s], 82, 0, 82, 86, Yes, Yes], [[a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a3, s], 82, 0, 82, 86, Yes, Yes], [[a1, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a3, s], 82, 0, 82, 86, Yes, Yes], [[a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a1, a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a2, a3, s], 82, 0, 82, 86, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69ad9936b29b89f0939351f64425fb22d190bd9c8b40af00d2e088a0945d9380","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155045,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25], candidate flows=[20, 20, 21, 21, 22, 22, 23, 23, 24, 24]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(110, 110, [[[s], 110, 0, 110, 115, Yes, Yes], [[a0, s], 110, 0, 110, 115, Yes, Yes], [[a1, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, s], 110, 0, 110, 115, Yes, Yes], [[a2, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, s], 110, 0, 110, 115, Yes, Yes], [[a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a3, s], 110, 0, 110, 115, Yes, Yes], [[a1, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a3, s], 110, 0, 110, 115, Yes, Yes], [[a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a4, s], 110, 0, 110, 115, Yes, Yes], [[a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (110, 110, [[[s], 110, 0, 110, 115, Yes, Yes], [[a0, s], 110, 0, 110, 115, Yes, Yes], [[a1, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, s], 110, 0, 110, 115, Yes, Yes], [[a2, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, s], 110, 0, 110, 115, Yes, Yes], [[a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a3, s], 110, 0, 110, 115, Yes, Yes], [[a1, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a3, s], 110, 0, 110, 115, Yes, Yes], [[a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a4, s], 110, 0, 110, 115, Yes, Yes], [[a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"560caa9af7419f46ab0b983587f8584a3d0e8b8599a52c2d5941943f30197f9c","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155046,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[22, 22, 23, 23], candidate flows=[21, 21, 22, 22]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(43, 43, [[[s], 43, 0, 43, 45, Yes, Yes], [[a0, s], 43, 0, 43, 45, Yes, Yes], [[a1, s], 43, 0, 43, 45, Yes, Yes], [[a0, a1, s], 43, 0, 43, 45, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (43, 43, [[[s], 43, 0, 43, 45, Yes, Yes], [[a0, s], 43, 0, 43, 45, Yes, Yes], [[a1, s], 43, 0, 43, 45, Yes, Yes], [[a0, a1, s], 43, 0, 43, 45, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef177af067c0cf4be6d7248ae7bdeb5b35c77ffbf1deafab93cf4127328c41a8","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155047,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[23, 23, 24, 24, 25, 25, 28], candidate flows=[22, 22, 23, 23, 24, 24, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(69, 69, [[[s], 69, 0, 69, 72, Yes, Yes], [[a0, s], 69, 0, 69, 72, Yes, Yes], [[a1, s], 69, 0, 69, 72, Yes, Yes], [[a0, a1, s], 69, 0, 69, 72, Yes, Yes], [[a2, s], 69, 0, 69, 72, Yes, Yes], [[a0, a2, s], 69, 0, 69, 72, Yes, Yes], [[a1, a2, s], 69, 0, 69, 72, Yes, Yes], [[a0, a1, a2, s], 69, 0, 69, 72, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (69, 69, [[[s], 69, 0, 69, 72, Yes, Yes], [[a0, s], 69, 0, 69, 72, Yes, Yes], [[a1, s], 69, 0, 69, 72, Yes, Yes], [[a0, a1, s], 69, 0, 69, 72, Yes, Yes], [[a2, s], 69, 0, 69, 72, Yes, Yes], [[a0, a2, s], 69, 0, 69, 72, Yes, Yes], [[a1, a2, s], 69, 0, 69, 72, Yes, Yes], [[a0, a1, a2, s], 69, 0, 69, 72, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a40ccc7a68e634c3a68b6c8dee15d91ab3836b5b5a36f32860c9d5ac4c2d75e0","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155048,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[24, 24, 25, 25, 26, 26, 27, 27], candidate flows=[23, 23, 24, 24, 25, 25, 26, 26]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(98, 98, [[[s], 98, 0, 98, 102, Yes, Yes], [[a0, s], 98, 0, 98, 102, Yes, Yes], [[a1, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, s], 98, 0, 98, 102, Yes, Yes], [[a2, s], 98, 0, 98, 102, Yes, Yes], [[a0, a2, s], 98, 0, 98, 102, Yes, Yes], [[a1, a2, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a2, s], 98, 0, 98, 102, Yes, Yes], [[a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a3, s], 98, 0, 98, 102, Yes, Yes], [[a1, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a3, s], 98, 0, 98, 102, Yes, Yes], [[a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a1, a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a2, a3, s], 98, 0, 98, 102, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (98, 98, [[[s], 98, 0, 98, 102, Yes, Yes], [[a0, s], 98, 0, 98, 102, Yes, Yes], [[a1, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, s], 98, 0, 98, 102, Yes, Yes], [[a2, s], 98, 0, 98, 102, Yes, Yes], [[a0, a2, s], 98, 0, 98, 102, Yes, Yes], [[a1, a2, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a2, s], 98, 0, 98, 102, Yes, Yes], [[a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a3, s], 98, 0, 98, 102, Yes, Yes], [[a1, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a3, s], 98, 0, 98, 102, Yes, Yes], [[a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a1, a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a2, a3, s], 98, 0, 98, 102, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f81f6bf2d48b4ba47a44d7678de695fb4a4cbb97c0df42e3168eeb2bbfe8d5a1","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155049,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29], candidate flows=[24, 24, 25, 25, 26, 26, 27, 27, 28, 28]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(130, 130, [[[s], 130, 0, 130, 135, Yes, Yes], [[a0, s], 130, 0, 130, 135, Yes, Yes], [[a1, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, s], 130, 0, 130, 135, Yes, Yes], [[a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (130, 130, [[[s], 130, 0, 130, 135, Yes, Yes], [[a0, s], 130, 0, 130, 135, Yes, Yes], [[a1, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, s], 130, 0, 130, 135, Yes, Yes], [[a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72ff24c3afb61f7a7b8841fd300b3d0a1a8a75a8f4d05d2b9dac428d5ca75ebf","family_id":"qf-mod46-mfmc-flow-cut-upper-bound","seed":202611155050,"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\"]}"},"fields":{"title":"A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[26, 26, 27, 27, 31], candidate flows=[25, 25, 26, 26, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).","answer":{"kind":"expression","value":"(51, 51, [[[s], 51, 0, 51, 53, Yes, Yes], [[a0, s], 51, 0, 51, 53, Yes, Yes], [[a1, s], 51, 0, 51, 53, Yes, Yes], [[a0, a1, s], 51, 0, 51, 53, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)","explanation":"Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (51, 51, [[[s], 51, 0, 51, 53, Yes, Yes], [[a0, s], 51, 0, 51, 53, Yes, Yes], [[a1, s], 51, 0, 51, 53, Yes, Yes], [[a0, a1, s], 51, 0, 51, 53, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)."},"skills":["A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb1a780ba7121af49134d33d63bb8b2bd96c12a490f067152b063b7ed22c3bd4","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155051,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[2, 2, 3, 3, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bbdb94b8ed428b5dcad3b4c6ecc7e2a4144d95f6cb38f134a478d7a57ac2620","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155052,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[3, 3, 4, 4, 5, 5]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4f373fa8a8a98679fe0a08e9d08264268de54a91924474cdd10da8d02c0b0cf","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155053,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[4, 4, 5, 5, 6, 6, 7, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1634c353b8d1a717014bb01960be00e02c5bad471b7e54b975bb53835d87066b","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155054,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10e3ab641a135050524f763d08886b27b16308b01d50c6eb4e8a5e4fbbf03f3f","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155055,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[6, 6, 7, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18ca294c62f69f8892171c5a5aa159ef6f4c5971e2b9718358a484bf35c4802f","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155056,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[7, 7, 8, 8, 9, 9]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42f34abc217ca15cac8f7049075885509f8b1026225c49b76987da20b302ec1a","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155057,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[8, 8, 9, 9, 10, 10, 11, 11, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d81e9aa3fb3298399c1ba43e4bb23461aa5995ca594dd404d835edbfdb7afd62","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155058,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3acd9b8eb99ac2a5e2ed033d918905578fdc560ba0b600f1a1a5070bb2f3782d","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155059,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[10, 10, 11, 11]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c479246467f68badf4ecca055e9f3d7b4ed3c2808da45db573e68e65526860c","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155060,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[11, 11, 12, 12, 13, 13, 16]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e663d2751a8814bb1260740ec6832ddc1491be5ef58745e36d0013efe2636819","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155061,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[12, 12, 13, 13, 14, 14, 15, 15]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12c256257ac27f8edcf2f16a0b07efc1136606bfe7834689f12bd03de5161053","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155062,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b9f526ff0857202b41949b60b99f463700445a77871d3501b70904cb268220a","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155063,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[14, 14, 15, 15, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28a8ee97dc5ae9f1fe107316c66774b3fc26ad1bf5d8397756818fd23b2bf885","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155064,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64f91278fc511be36be07b7592c1c8d24a46844705fc0d3302800ba9a9e8888c","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155065,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[16, 16, 17, 17, 18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f844a3f98084ec8cbc70839af28961e169ef7b413cf5cd2f00626709cea15d57","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155066,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af56480d2392e59984cc51c4f75eb044cd0f167892ce344ff48cfec566ead85b","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155067,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fce9c66636bb4e8997efa426ec3f67e85721a502251780f07214e3c8a0fc957","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155068,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[19, 19, 20, 20, 21, 21]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1467b24c1d05a671735bb963403e6b987ec49c0074d186912510466166ce3b14","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155069,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[20, 20, 21, 21, 22, 22, 23, 23, 25]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73ad7a3bd3d067ca20dc10ed026803248eae38ccafb8cc954352bab453d5f474","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155070,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0aa6d2a9230436631e1b9fe0555f6e34687c4ea0fd312f777a8fc8cd9f4f107","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155071,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[22, 22, 23, 23]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df64619c2133e124a4f00d8665fb21d4e15bef266936e4cc70a36bdca1b02064","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155072,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[23, 23, 24, 24, 25, 25, 28]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fd7cb6d4833ba42a4ab6c36140905760203250178a3245238e486dec88e0a41","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155073,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[24, 24, 25, 25, 26, 26, 27, 27]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06293937362e10160c212cc3ea22f858bb6db1a0ef9c40d390632ae7fd476037","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155074,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02b434d08b49138da6af6e6ab3a352fe267e29b8dbaa67582177b97b7c906df0","family_id":"qf-mod46-mfmc-minimum-cut-audit","seed":202611155075,"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\"]}"},"fields":{"title":"The Minimum Cut Is Selected from the Complete Cut Family","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[26, 26, 27, 27, 31]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).","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":"Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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)."},"skills":["The Minimum Cut Is Selected from the Complete Cut Family"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2f40892f70dfadf9930bbf500a9e5537fd65ec9fcbfe11bb3e9c83853afedf3","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155076,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[2, 2, 3, 3, 7], candidate flows=[2, 2, 3, 3, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","answer":{"kind":"expression","value":"(5, 5, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (5, 5, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3289aaa4338d03f6eb747c10fb9baed3e5519e12bc073f0bfffaef637dbc2ad5","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155077,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[3, 3, 4, 4, 5, 5], candidate flows=[3, 3, 4, 4, 5, 5]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"090be1577566064b03615d5eb03084de013f9f8076f9c3a0e70ae269857cb786","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155078,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[4, 4, 5, 5, 6, 6, 7, 7], candidate flows=[4, 4, 5, 5, 6, 6, 7, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ab8653e92550d12e5263816086b31f90b2a4caf39c8e5af0a217665ff63b4d8","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155079,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10], candidate flows=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4e64dea0b658a8eafe671f771ffa2e243418a047b2dcb4f9fb8aa3c92f1f16c","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155080,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[6, 6, 7, 7], candidate flows=[6, 6, 7, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","answer":{"kind":"expression","value":"(13, 13, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (13, 13, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be8652a91bb70c0436b98e00a3b10222d751d29d5dbd8fd5bcfcd83a3eaef498","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155081,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[7, 7, 8, 8, 9, 9], candidate flows=[7, 7, 8, 8, 9, 9]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b4aacd0033e9902058fe60496f5fb863d12517da797aaa0d3c0d79416c10c18","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155082,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[8, 8, 9, 9, 10, 10, 11, 11, 13], candidate flows=[8, 8, 9, 9, 10, 10, 11, 11, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11910861e66fbc7afcf620c2ef568be992f9a00a8bc871eaa4c1977adbc2a3e7","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155083,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13], candidate flows=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05b377df9c57e3bd0842c5123a083b5611208dbf919f6f22d588a4260ab582b5","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155084,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[10, 10, 11, 11], candidate flows=[10, 10, 11, 11]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","answer":{"kind":"expression","value":"(21, 21, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (21, 21, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b8d482b5d3e1af5ec1bc9bf19bd5009b1222da96a3ffcc32848ea1f75da7aca","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155085,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[11, 11, 12, 12, 13, 13, 16], candidate flows=[11, 11, 12, 12, 13, 13, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd49b878a025f7d0920decc2ef36d23bb03cbf60033ce34e2fcc4e075947aa8c","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155086,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[12, 12, 13, 13, 14, 14, 15, 15], candidate flows=[12, 12, 13, 13, 14, 14, 15, 15]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b713ad7f45c824acc3ab89406dd2e993dfb26446ce300da65a43c1bb67981924","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155087,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17], candidate flows=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1256df878729bbf7e88dedda3a0833949c1c8b0e199cd82c8d9b58c51d6c7914","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155088,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[14, 14, 15, 15, 19], candidate flows=[14, 14, 15, 15, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","answer":{"kind":"expression","value":"(29, 29, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (29, 29, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd84887d9d08fc6ede8abd7e10116a04f1b80f20aa9e73f280832aa7917541c7","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155089,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[15, 15, 16, 16, 17, 17], candidate flows=[15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffc185449614cc1aba76511c4f4ec6bd49f29d3b3b09bae5055d6cd0ca3985c6","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155090,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[16, 16, 17, 17, 18, 18, 19, 19], candidate flows=[16, 16, 17, 17, 18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f451f09b19e261fb5799d47d6c4dcae12bb49bbfc3aa4ca0d01e27a96cb7822","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155091,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22], candidate flows=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"958280471eaae6564a585153b204fde6f440549b3fd3b269e7d6d6636d2fea47","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155092,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[18, 18, 19, 19], candidate flows=[18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","answer":{"kind":"expression","value":"(37, 37, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (37, 37, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b301814fc1a67f2c01341304b501d682013026bce2b91dc47f316cc2cd580ef5","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155093,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[19, 19, 20, 20, 21, 21], candidate flows=[19, 19, 20, 20, 21, 21]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83d8942391ec70eb05bde0a43d22ce9be944f5e923c0881a5d9494368a214d66","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155094,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[20, 20, 21, 21, 22, 22, 23, 23, 25], candidate flows=[20, 20, 21, 21, 22, 22, 23, 23, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f96f241f047cab8bcf03433df99a78e5aaa896e799a16db83249a3eac8ec8ead","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155095,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25], candidate flows=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d4671358c3e3ef164d57c251f922633e61f36cec15c51069e2545d3a112fc69","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155096,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[22, 22, 23, 23], candidate flows=[22, 22, 23, 23]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","answer":{"kind":"expression","value":"(45, 45, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (45, 45, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adf4a0557111dfa0423f0662be0b504fd1d8d7db8e2417a5287b29c94fd1b976","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155097,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[23, 23, 24, 24, 25, 25, 28], candidate flows=[23, 23, 24, 24, 25, 25, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adf7148cd4b92afa4cbde6aba45a7bd57daf1d9a1777848ee59a09ebedc4562e","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155098,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[24, 24, 25, 25, 26, 26, 27, 27], candidate flows=[24, 24, 25, 25, 26, 26, 27, 27]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2b1258d5fb17bc7828a0d868fc85746e779f9d74fd712f388f0b1a39d9c0d96","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155099,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered 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, aligned nonnegative exact capacities=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29], candidate flows=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_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","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1be3cf99054dcfe5f589341c1d502900394221685c2519ba34e0f7aa87483bfc","family_id":"qf-mod46-mfmc-equal-value-optimality","seed":202611155100,"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\"]}"},"fields":{"title":"An Equal-Valued Feasible Flow and Cut Certify Both Optima","stem":"Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[26, 26, 27, 27, 31], candidate flows=[26, 26, 27, 27, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).","answer":{"kind":"expression","value":"(53, 53, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)","explanation":"First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (53, 53, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)."},"skills":["An Equal-Valued Feasible Flow and Cut Certify Both Optima"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}