{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r07-finite-flow-network-0001","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ede38bc1b5ee5fc7f4e7720ac2abd5462605ae725682495bce0391f5618a074","family_id":"qf-mod44-flow-network-qualification","seed":202611137001,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[5,1],[5,1],[5,1],[5,1]],\"edges\":[[\"a0\",\"s0\"],[\"a0\",\"t0\"],[\"s0\",\"a0\"],[\"t0\",\"a0\"]],\"flows\":[[2,1],[3,1],[4,1],[1,1]],\"sink\":\"t0\",\"source\":\"s0\",\"vertices\":[\"a0\",\"s0\",\"t0\",\"u0\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s0, a0, t0, u0], ordered directed edges=[[s0, a0], [a0, s0], [a0, t0], [t0, a0]], distinct source=s0, sink=t0, aligned capacities=[5, 5, 5, 5] and candidate flows=[4, 2, 3, 1]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a0, s0, t0, u0], s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, 4], [[t0, a0], 5, 1]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a0, s0, t0, u0], s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, 4], [[t0, a0], 5, 1]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0002","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e4555947edb33c281a31f491e1ded81ee2af51edc824b821c83f06aa28507c4","family_id":"qf-mod44-flow-network-qualification","seed":202611137002,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[10,1],[10,1],[10,1],[10,1]],\"edges\":[[\"a1\",\"s1\"],[\"a1\",\"t1\"],[\"s1\",\"a1\"],[\"t1\",\"a1\"]],\"flows\":[[4,1],[6,1],[8,1],[2,1]],\"sink\":\"t1\",\"source\":\"s1\",\"vertices\":[\"a1\",\"s1\",\"t1\",\"u1\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s1, a1, t1, u1], ordered directed edges=[[s1, a1], [a1, s1], [a1, t1], [t1, a1]], distinct source=s1, sink=t1, aligned capacities=[10, 10, 10, 10] and candidate flows=[8, 4, 6, 2]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a1, s1, t1, u1], s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a1, s1, t1, u1], s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0003","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b94c6f188dab3cad378bb540764dfd8732144ecf57e6928935a198b93bc645fe","family_id":"qf-mod44-flow-network-qualification","seed":202611137003,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[15,1],[15,1],[15,1],[15,1]],\"edges\":[[\"a2\",\"s2\"],[\"a2\",\"t2\"],[\"s2\",\"a2\"],[\"t2\",\"a2\"]],\"flows\":[[6,1],[9,1],[12,1],[3,1]],\"sink\":\"t2\",\"source\":\"s2\",\"vertices\":[\"a2\",\"s2\",\"t2\",\"u2\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s2, a2, t2, u2], ordered directed edges=[[s2, a2], [a2, s2], [a2, t2], [t2, a2]], distinct source=s2, sink=t2, aligned capacities=[15, 15, 15, 15] and candidate flows=[12, 6, 9, 3]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a2, s2, t2, u2], s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a2, s2, t2, u2], s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0004","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9d57da2068b97ee19cf81be118b5795e8b340138bf10a14e01ece4736381bda","family_id":"qf-mod44-flow-network-qualification","seed":202611137004,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[20,1],[20,1],[20,1],[20,1]],\"edges\":[[\"a3\",\"s3\"],[\"a3\",\"t3\"],[\"s3\",\"a3\"],[\"t3\",\"a3\"]],\"flows\":[[8,1],[12,1],[16,1],[4,1]],\"sink\":\"t3\",\"source\":\"s3\",\"vertices\":[\"a3\",\"s3\",\"t3\",\"u3\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s3, a3, t3, u3], ordered directed edges=[[s3, a3], [a3, s3], [a3, t3], [t3, a3]], distinct source=s3, sink=t3, aligned capacities=[20, 20, 20, 20] and candidate flows=[16, 8, 12, 4]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a3, s3, t3, u3], s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, 16], [[t3, a3], 20, 4]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a3, s3, t3, u3], s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, 16], [[t3, a3], 20, 4]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0005","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c22254626185e942d3186573cd9f1c94c2676a598569e167e876e9ec7e5d23b","family_id":"qf-mod44-flow-network-qualification","seed":202611137005,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[25,1],[25,1],[25,1],[25,1]],\"edges\":[[\"a4\",\"s4\"],[\"a4\",\"t4\"],[\"s4\",\"a4\"],[\"t4\",\"a4\"]],\"flows\":[[10,1],[15,1],[20,1],[5,1]],\"sink\":\"t4\",\"source\":\"s4\",\"vertices\":[\"a4\",\"s4\",\"t4\",\"u4\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s4, a4, t4, u4], ordered directed edges=[[s4, a4], [a4, s4], [a4, t4], [t4, a4]], distinct source=s4, sink=t4, aligned capacities=[25, 25, 25, 25] and candidate flows=[20, 10, 15, 5]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a4, s4, t4, u4], s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 20], [[t4, a4], 25, 5]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a4, s4, t4, u4], s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 20], [[t4, a4], 25, 5]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0006","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de69327a78228e77f0c247788f0c90aa4ea1822d1419c6bcd86c1761ed8be300","family_id":"qf-mod44-flow-network-qualification","seed":202611137006,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[30,1],[30,1],[30,1],[30,1]],\"edges\":[[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"s5\",\"a5\"],[\"t5\",\"a5\"]],\"flows\":[[12,1],[18,1],[24,1],[6,1]],\"sink\":\"t5\",\"source\":\"s5\",\"vertices\":[\"a5\",\"s5\",\"t5\",\"u5\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s5, a5, t5, u5], ordered directed edges=[[s5, a5], [a5, s5], [a5, t5], [t5, a5]], distinct source=s5, sink=t5, aligned capacities=[30, 30, 30, 30] and candidate flows=[24, 12, 18, 6]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a5, s5, t5, u5], s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a5, s5, t5, u5], s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0007","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40f636a6b5ef02b8518dbf9c279ab2990c542b701deb9a383905de10d68ed43c","family_id":"qf-mod44-flow-network-qualification","seed":202611137007,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[35,1],[35,1],[35,1],[35,1]],\"edges\":[[\"a6\",\"s6\"],[\"a6\",\"t6\"],[\"s6\",\"a6\"],[\"t6\",\"a6\"]],\"flows\":[[14,1],[21,1],[28,1],[7,1]],\"sink\":\"t6\",\"source\":\"s6\",\"vertices\":[\"a6\",\"s6\",\"t6\",\"u6\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s6, a6, t6, u6], ordered directed edges=[[s6, a6], [a6, s6], [a6, t6], [t6, a6]], distinct source=s6, sink=t6, aligned capacities=[35, 35, 35, 35] and candidate flows=[28, 14, 21, 7]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a6, s6, t6, u6], s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a6, s6, t6, u6], s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0008","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af67c0c7d585c4111416b8d8144bf4904c5a6e72ee8418d5c65fe2d3e781baa8","family_id":"qf-mod44-flow-network-qualification","seed":202611137008,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[40,1],[40,1],[40,1],[40,1]],\"edges\":[[\"a7\",\"s7\"],[\"a7\",\"t7\"],[\"s7\",\"a7\"],[\"t7\",\"a7\"]],\"flows\":[[16,1],[24,1],[32,1],[8,1]],\"sink\":\"t7\",\"source\":\"s7\",\"vertices\":[\"a7\",\"s7\",\"t7\",\"u7\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s7, a7, t7, u7], ordered directed edges=[[s7, a7], [a7, s7], [a7, t7], [t7, a7]], distinct source=s7, sink=t7, aligned capacities=[40, 40, 40, 40] and candidate flows=[32, 16, 24, 8]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a7, s7, t7, u7], s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 24], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a7, s7, t7, u7], s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 24], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0009","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8974c47184ba52b15eb5f9cd5abf548dd00c55cf231c5591ffb1ff7afde8854a","family_id":"qf-mod44-flow-network-qualification","seed":202611137009,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[45,1],[45,1],[45,1],[45,1]],\"edges\":[[\"a8\",\"s8\"],[\"a8\",\"t8\"],[\"s8\",\"a8\"],[\"t8\",\"a8\"]],\"flows\":[[18,1],[27,1],[36,1],[9,1]],\"sink\":\"t8\",\"source\":\"s8\",\"vertices\":[\"a8\",\"s8\",\"t8\",\"u8\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s8, a8, t8, u8], ordered directed edges=[[s8, a8], [a8, s8], [a8, t8], [t8, a8]], distinct source=s8, sink=t8, aligned capacities=[45, 45, 45, 45] and candidate flows=[36, 18, 27, 9]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a8, s8, t8, u8], s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a8, s8, t8, u8], s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0010","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7248a73f5611ddbcd577ead689945f4bd5c9e7ded6032f17806acc13cfd98a0","family_id":"qf-mod44-flow-network-qualification","seed":202611137010,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[50,1],[50,1],[50,1],[50,1]],\"edges\":[[\"a9\",\"s9\"],[\"a9\",\"t9\"],[\"s9\",\"a9\"],[\"t9\",\"a9\"]],\"flows\":[[20,1],[30,1],[40,1],[10,1]],\"sink\":\"t9\",\"source\":\"s9\",\"vertices\":[\"a9\",\"s9\",\"t9\",\"u9\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s9, a9, t9, u9], ordered directed edges=[[s9, a9], [a9, s9], [a9, t9], [t9, a9]], distinct source=s9, sink=t9, aligned capacities=[50, 50, 50, 50] and candidate flows=[40, 20, 30, 10]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a9, s9, t9, u9], s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, 40], [[t9, a9], 50, 10]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a9, s9, t9, u9], s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, 40], [[t9, a9], 50, 10]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0011","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24b72f29c5123acc58bc3ea554f06b3c28c120dc91517339d3d784de77fae656","family_id":"qf-mod44-flow-network-qualification","seed":202611137011,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[55,1],[55,1],[55,1],[55,1]],\"edges\":[[\"a10\",\"s10\"],[\"a10\",\"t10\"],[\"s10\",\"a10\"],[\"t10\",\"a10\"]],\"flows\":[[22,1],[33,1],[44,1],[11,1]],\"sink\":\"t10\",\"source\":\"s10\",\"vertices\":[\"a10\",\"s10\",\"t10\",\"u10\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s10, a10, t10, u10], ordered directed edges=[[s10, a10], [a10, s10], [a10, t10], [t10, a10]], distinct source=s10, sink=t10, aligned capacities=[55, 55, 55, 55] and candidate flows=[44, 22, 33, 11]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a10, s10, t10, u10], s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 44], [[t10, a10], 55, 11]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a10, s10, t10, u10], s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 44], [[t10, a10], 55, 11]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0012","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"782a17c241eac3ab6c5f77763b86481f86f9e1d4c3db6ce9f16122f22cda3027","family_id":"qf-mod44-flow-network-qualification","seed":202611137012,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[60,1],[60,1],[60,1],[60,1]],\"edges\":[[\"a11\",\"s11\"],[\"a11\",\"t11\"],[\"s11\",\"a11\"],[\"t11\",\"a11\"]],\"flows\":[[24,1],[36,1],[48,1],[12,1]],\"sink\":\"t11\",\"source\":\"s11\",\"vertices\":[\"a11\",\"s11\",\"t11\",\"u11\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s11, a11, t11, u11], ordered directed edges=[[s11, a11], [a11, s11], [a11, t11], [t11, a11]], distinct source=s11, sink=t11, aligned capacities=[60, 60, 60, 60] and candidate flows=[48, 24, 36, 12]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a11, s11, t11, u11], s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 36], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a11, s11, t11, u11], s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 36], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0013","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56d5a5bc19d55a3082b07394faea0fab3c93caff809145408b4a11b63c6f2711","family_id":"qf-mod44-flow-network-qualification","seed":202611137013,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[65,1],[65,1],[65,1],[65,1]],\"edges\":[[\"a12\",\"s12\"],[\"a12\",\"t12\"],[\"s12\",\"a12\"],[\"t12\",\"a12\"]],\"flows\":[[26,1],[39,1],[52,1],[13,1]],\"sink\":\"t12\",\"source\":\"s12\",\"vertices\":[\"a12\",\"s12\",\"t12\",\"u12\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s12, a12, t12, u12], ordered directed edges=[[s12, a12], [a12, s12], [a12, t12], [t12, a12]], distinct source=s12, sink=t12, aligned capacities=[65, 65, 65, 65] and candidate flows=[52, 26, 39, 13]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a12, s12, t12, u12], s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, 52], [[t12, a12], 65, 13]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a12, s12, t12, u12], s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, 52], [[t12, a12], 65, 13]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0014","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce7d53fba5944509648c794303815f391bdf5b5c2efb9912dc24241e89de48c0","family_id":"qf-mod44-flow-network-qualification","seed":202611137014,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[70,1],[70,1],[70,1],[70,1]],\"edges\":[[\"a13\",\"s13\"],[\"a13\",\"t13\"],[\"s13\",\"a13\"],[\"t13\",\"a13\"]],\"flows\":[[28,1],[42,1],[56,1],[14,1]],\"sink\":\"t13\",\"source\":\"s13\",\"vertices\":[\"a13\",\"s13\",\"t13\",\"u13\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s13, a13, t13, u13], ordered directed edges=[[s13, a13], [a13, s13], [a13, t13], [t13, a13]], distinct source=s13, sink=t13, aligned capacities=[70, 70, 70, 70] and candidate flows=[56, 28, 42, 14]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a13, s13, t13, u13], s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 42], [[s13, a13], 70, 56], [[t13, a13], 70, 14]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a13, s13, t13, u13], s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 42], [[s13, a13], 70, 56], [[t13, a13], 70, 14]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0015","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8de8562abf9fed720cde4d4139af612b578735bd676a02830d33d3f4330151ec","family_id":"qf-mod44-flow-network-qualification","seed":202611137015,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[75,1],[75,1],[75,1],[75,1]],\"edges\":[[\"a14\",\"s14\"],[\"a14\",\"t14\"],[\"s14\",\"a14\"],[\"t14\",\"a14\"]],\"flows\":[[30,1],[45,1],[60,1],[15,1]],\"sink\":\"t14\",\"source\":\"s14\",\"vertices\":[\"a14\",\"s14\",\"t14\",\"u14\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s14, a14, t14, u14], ordered directed edges=[[s14, a14], [a14, s14], [a14, t14], [t14, a14]], distinct source=s14, sink=t14, aligned capacities=[75, 75, 75, 75] and candidate flows=[60, 30, 45, 15]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a14, s14, t14, u14], s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a14, s14, t14, u14], s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0016","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d29bfaaebe35a409c67a293774df7107c7efe7e0fd4e982a26ee5ab0c8c0e620","family_id":"qf-mod44-flow-network-qualification","seed":202611137016,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[80,1],[80,1],[80,1],[80,1]],\"edges\":[[\"a15\",\"s15\"],[\"a15\",\"t15\"],[\"s15\",\"a15\"],[\"t15\",\"a15\"]],\"flows\":[[32,1],[48,1],[64,1],[16,1]],\"sink\":\"t15\",\"source\":\"s15\",\"vertices\":[\"a15\",\"s15\",\"t15\",\"u15\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s15, a15, t15, u15], ordered directed edges=[[s15, a15], [a15, s15], [a15, t15], [t15, a15]], distinct source=s15, sink=t15, aligned capacities=[80, 80, 80, 80] and candidate flows=[64, 32, 48, 16]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a15, s15, t15, u15], s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, 64], [[t15, a15], 80, 16]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a15, s15, t15, u15], s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, 64], [[t15, a15], 80, 16]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0017","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c91a3c83d0dbbad47d26dbf8965e791331cf2febf57126f623b5862a1c9c839","family_id":"qf-mod44-flow-network-qualification","seed":202611137017,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[85,1],[85,1],[85,1],[85,1]],\"edges\":[[\"a16\",\"s16\"],[\"a16\",\"t16\"],[\"s16\",\"a16\"],[\"t16\",\"a16\"]],\"flows\":[[34,1],[51,1],[68,1],[17,1]],\"sink\":\"t16\",\"source\":\"s16\",\"vertices\":[\"a16\",\"s16\",\"t16\",\"u16\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s16, a16, t16, u16], ordered directed edges=[[s16, a16], [a16, s16], [a16, t16], [t16, a16]], distinct source=s16, sink=t16, aligned capacities=[85, 85, 85, 85] and candidate flows=[68, 34, 51, 17]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a16, s16, t16, u16], s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 68], [[t16, a16], 85, 17]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a16, s16, t16, u16], s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 68], [[t16, a16], 85, 17]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0018","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76865c5aa67496b0f1ea6ad82aa9f7389554d60723b33e16ff3199e040dbf8d4","family_id":"qf-mod44-flow-network-qualification","seed":202611137018,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[90,1],[90,1],[90,1],[90,1]],\"edges\":[[\"a17\",\"s17\"],[\"a17\",\"t17\"],[\"s17\",\"a17\"],[\"t17\",\"a17\"]],\"flows\":[[36,1],[54,1],[72,1],[18,1]],\"sink\":\"t17\",\"source\":\"s17\",\"vertices\":[\"a17\",\"s17\",\"t17\",\"u17\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s17, a17, t17, u17], ordered directed edges=[[s17, a17], [a17, s17], [a17, t17], [t17, a17]], distinct source=s17, sink=t17, aligned capacities=[90, 90, 90, 90] and candidate flows=[72, 36, 54, 18]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a17, s17, t17, u17], s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 54], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a17, s17, t17, u17], s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 54], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0019","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2941b2f8e17c0b87d4d5e890084358c6ea614b4c832d8a13005f5782af55002","family_id":"qf-mod44-flow-network-qualification","seed":202611137019,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[95,1],[95,1],[95,1],[95,1]],\"edges\":[[\"a18\",\"s18\"],[\"a18\",\"t18\"],[\"s18\",\"a18\"],[\"t18\",\"a18\"]],\"flows\":[[38,1],[57,1],[76,1],[19,1]],\"sink\":\"t18\",\"source\":\"s18\",\"vertices\":[\"a18\",\"s18\",\"t18\",\"u18\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s18, a18, t18, u18], ordered directed edges=[[s18, a18], [a18, s18], [a18, t18], [t18, a18]], distinct source=s18, sink=t18, aligned capacities=[95, 95, 95, 95] and candidate flows=[76, 38, 57, 19]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a18, s18, t18, u18], s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, 76], [[t18, a18], 95, 19]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a18, s18, t18, u18], s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, 76], [[t18, a18], 95, 19]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0020","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b3b742b25b6812c3266c441e4c2389359897147f37111c26158e5f2605db2a2","family_id":"qf-mod44-flow-network-qualification","seed":202611137020,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[100,1],[100,1],[100,1],[100,1]],\"edges\":[[\"a19\",\"s19\"],[\"a19\",\"t19\"],[\"s19\",\"a19\"],[\"t19\",\"a19\"]],\"flows\":[[40,1],[60,1],[80,1],[20,1]],\"sink\":\"t19\",\"source\":\"s19\",\"vertices\":[\"a19\",\"s19\",\"t19\",\"u19\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s19, a19, t19, u19], ordered directed edges=[[s19, a19], [a19, s19], [a19, t19], [t19, a19]], distinct source=s19, sink=t19, aligned capacities=[100, 100, 100, 100] and candidate flows=[80, 40, 60, 20]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a19, s19, t19, u19], s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 60], [[s19, a19], 100, 80], [[t19, a19], 100, 20]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a19, s19, t19, u19], s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 60], [[s19, a19], 100, 80], [[t19, a19], 100, 20]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0021","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7028b796bd0dfd5f54a507140e99c29ee18d5994abade94a8361b7610e2815e6","family_id":"qf-mod44-flow-network-qualification","seed":202611137021,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[105,1],[105,1],[105,1],[105,1]],\"edges\":[[\"a20\",\"s20\"],[\"a20\",\"t20\"],[\"s20\",\"a20\"],[\"t20\",\"a20\"]],\"flows\":[[42,1],[63,1],[84,1],[21,1]],\"sink\":\"t20\",\"source\":\"s20\",\"vertices\":[\"a20\",\"s20\",\"t20\",\"u20\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s20, a20, t20, u20], ordered directed edges=[[s20, a20], [a20, s20], [a20, t20], [t20, a20]], distinct source=s20, sink=t20, aligned capacities=[105, 105, 105, 105] and candidate flows=[84, 42, 63, 21]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a20, s20, t20, u20], s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a20, s20, t20, u20], s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0022","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b42f8a12fc2976c00ec675a69e19f455c8428ed85b3be46a5671f1ccab76d219","family_id":"qf-mod44-flow-network-qualification","seed":202611137022,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[110,1],[110,1],[110,1],[110,1]],\"edges\":[[\"a21\",\"s21\"],[\"a21\",\"t21\"],[\"s21\",\"a21\"],[\"t21\",\"a21\"]],\"flows\":[[44,1],[66,1],[88,1],[22,1]],\"sink\":\"t21\",\"source\":\"s21\",\"vertices\":[\"a21\",\"s21\",\"t21\",\"u21\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s21, a21, t21, u21], ordered directed edges=[[s21, a21], [a21, s21], [a21, t21], [t21, a21]], distinct source=s21, sink=t21, aligned capacities=[110, 110, 110, 110] and candidate flows=[88, 44, 66, 22]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a21, s21, t21, u21], s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 66], [[s21, a21], 110, 88], [[t21, a21], 110, 22]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a21, s21, t21, u21], s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 66], [[s21, a21], 110, 88], [[t21, a21], 110, 22]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0023","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a347d352f05217c499ff206e8f3d5e369c0b7bd96f83b82238f554460a7ee266","family_id":"qf-mod44-flow-network-qualification","seed":202611137023,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[115,1],[115,1],[115,1],[115,1]],\"edges\":[[\"a22\",\"s22\"],[\"a22\",\"t22\"],[\"s22\",\"a22\"],[\"t22\",\"a22\"]],\"flows\":[[46,1],[69,1],[92,1],[23,1]],\"sink\":\"t22\",\"source\":\"s22\",\"vertices\":[\"a22\",\"s22\",\"t22\",\"u22\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s22, a22, t22, u22], ordered directed edges=[[s22, a22], [a22, s22], [a22, t22], [t22, a22]], distinct source=s22, sink=t22, aligned capacities=[115, 115, 115, 115] and candidate flows=[92, 46, 69, 23]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a22, s22, t22, u22], s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a22, s22, t22, u22], s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0024","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be84d8b504b63b5293a2403c7469ce8d027d1bbc0ac55045960470e4fbfdeab6","family_id":"qf-mod44-flow-network-qualification","seed":202611137024,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[120,1],[120,1],[120,1],[120,1]],\"edges\":[[\"a23\",\"s23\"],[\"a23\",\"t23\"],[\"s23\",\"a23\"],[\"t23\",\"a23\"]],\"flows\":[[48,1],[72,1],[96,1],[24,1]],\"sink\":\"t23\",\"source\":\"s23\",\"vertices\":[\"a23\",\"s23\",\"t23\",\"u23\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s23, a23, t23, u23], ordered directed edges=[[s23, a23], [a23, s23], [a23, t23], [t23, a23]], distinct source=s23, sink=t23, aligned capacities=[120, 120, 120, 120] and candidate flows=[96, 48, 72, 24]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a23, s23, t23, u23], s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a23, s23, t23, u23], s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0025","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc2241e60a0a1a961363bd01a2191c5f5f92dab278b87f40180d0d319a3a8671","family_id":"qf-mod44-flow-network-qualification","seed":202611137025,"semantic_template":"qf-mod44-flow-network-qualification/{\"capacities\":[[125,1],[125,1],[125,1],[125,1]],\"edges\":[[\"a24\",\"s24\"],[\"a24\",\"t24\"],[\"s24\",\"a24\"],[\"t24\",\"a24\"]],\"flows\":[[50,1],[75,1],[100,1],[25,1]],\"sink\":\"t24\",\"source\":\"s24\",\"vertices\":[\"a24\",\"s24\",\"t24\",\"u24\"]}"},"fields":{"title":"Network Signature and Candidate-Flow Feasibility Are Separate Layers","stem":"Use vertices=[s24, a24, t24, u24], ordered directed edges=[[s24, a24], [a24, s24], [a24, t24], [t24, a24]], distinct source=s24, sink=t24, aligned capacities=[125, 125, 125, 125] and candidate flows=[100, 50, 75, 25]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention).","answer":{"kind":"expression","value":"([a24, s24, t24, u24], s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, 100], [[t24, a24], 125, 25]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)","explanation":"Report canonical [edge,capacity,candidate flow] rows, antiparallel pair count and network qualification. Zero capacities and disconnected vertices are legal. This flag is not a certificate that candidate flows meet capacity or conservation constraints. The exact result, retaining the requested order, is ([a24, s24, t24, u24], s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, 100], [[t24, a24], 125, 25]], 2, Yes, finite_loop_free_simple_digraph_with_explicit_antiparallel_convention_zero_capacities_and_disconnected_vertices_allowed)."},"skills":["Network Signature and Candidate-Flow Feasibility Are Separate Layers"],"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-20261113-r07-finite-flow-network-0026","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"816e0208e679259d85e3b0c03815148db61eb9d28efb8d8e1206f63b63bdfd0e","family_id":"qf-mod44-flow-capacity-audit","seed":202611137026,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[5,1],[5,1],[5,1],[5,1]],\"edges\":[[\"a0\",\"s0\"],[\"a0\",\"t0\"],[\"s0\",\"a0\"],[\"t0\",\"a0\"]],\"flows\":[[2,1],[3,1],[-1,1],[1,1]],\"sink\":\"t0\",\"source\":\"s0\",\"vertices\":[\"a0\",\"s0\",\"t0\",\"u0\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s0, a0, t0, u0], ordered directed edges=[[s0, a0], [a0, s0], [a0, t0], [t0, a0]], distinct source=s0, sink=t0, aligned capacities=[5, 5, 5, 5] and candidate flows=[-1, 2, 3, 1]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, -1], [[t0, a0], 5, 1]], [[[a0, s0], 2, 5, Yes], [[a0, t0], 3, 5, Yes], [[s0, a0], -1, 5, No], [[t0, a0], 1, 5, Yes]], No, [[s0, a0], -1, 5, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, -1], [[t0, a0], 5, 1]], [[[a0, s0], 2, 5, Yes], [[a0, t0], 3, 5, Yes], [[s0, a0], -1, 5, No], [[t0, a0], 1, 5, Yes]], No, [[s0, a0], -1, 5, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0027","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"043c9aa9c306ea42a3d014582d912e804ddfeeb83184cc1c4200a51fddfabe64","family_id":"qf-mod44-flow-capacity-audit","seed":202611137027,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[10,1],[10,1],[10,1],[10,1]],\"edges\":[[\"a1\",\"s1\"],[\"a1\",\"t1\"],[\"s1\",\"a1\"],[\"t1\",\"a1\"]],\"flows\":[[4,1],[6,1],[12,1],[2,1]],\"sink\":\"t1\",\"source\":\"s1\",\"vertices\":[\"a1\",\"s1\",\"t1\",\"u1\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s1, a1, t1, u1], ordered directed edges=[[s1, a1], [a1, s1], [a1, t1], [t1, a1]], distinct source=s1, sink=t1, aligned capacities=[10, 10, 10, 10] and candidate flows=[12, 4, 6, 2]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 12], [[t1, a1], 10, 2]], [[[a1, s1], 4, 10, Yes], [[a1, t1], 6, 10, Yes], [[s1, a1], 12, 10, No], [[t1, a1], 2, 10, Yes]], No, [[s1, a1], 12, 10, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 12], [[t1, a1], 10, 2]], [[[a1, s1], 4, 10, Yes], [[a1, t1], 6, 10, Yes], [[s1, a1], 12, 10, No], [[t1, a1], 2, 10, Yes]], No, [[s1, a1], 12, 10, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0028","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1be036d98e66d726f754e259b9729df7a9bb5aba8422ccd67709dc285e41cb16","family_id":"qf-mod44-flow-capacity-audit","seed":202611137028,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[15,1],[15,1],[15,1],[15,1]],\"edges\":[[\"a2\",\"s2\"],[\"a2\",\"t2\"],[\"s2\",\"a2\"],[\"t2\",\"a2\"]],\"flows\":[[6,1],[9,1],[12,1],[3,1]],\"sink\":\"t2\",\"source\":\"s2\",\"vertices\":[\"a2\",\"s2\",\"t2\",\"u2\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s2, a2, t2, u2], ordered directed edges=[[s2, a2], [a2, s2], [a2, t2], [t2, a2]], distinct source=s2, sink=t2, aligned capacities=[15, 15, 15, 15] and candidate flows=[12, 6, 9, 3]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], [[[a2, s2], 6, 15, Yes], [[a2, t2], 9, 15, Yes], [[s2, a2], 12, 15, Yes], [[t2, a2], 3, 15, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], [[[a2, s2], 6, 15, Yes], [[a2, t2], 9, 15, Yes], [[s2, a2], 12, 15, Yes], [[t2, a2], 3, 15, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0029","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c29c4ef812660675a46b8172a0f2493aacdf99d87eeabeb983b188f6b4958cc","family_id":"qf-mod44-flow-capacity-audit","seed":202611137029,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[20,1],[20,1],[20,1],[20,1]],\"edges\":[[\"a3\",\"s3\"],[\"a3\",\"t3\"],[\"s3\",\"a3\"],[\"t3\",\"a3\"]],\"flows\":[[8,1],[12,1],[-4,1],[4,1]],\"sink\":\"t3\",\"source\":\"s3\",\"vertices\":[\"a3\",\"s3\",\"t3\",\"u3\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s3, a3, t3, u3], ordered directed edges=[[s3, a3], [a3, s3], [a3, t3], [t3, a3]], distinct source=s3, sink=t3, aligned capacities=[20, 20, 20, 20] and candidate flows=[-4, 8, 12, 4]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, -4], [[t3, a3], 20, 4]], [[[a3, s3], 8, 20, Yes], [[a3, t3], 12, 20, Yes], [[s3, a3], -4, 20, No], [[t3, a3], 4, 20, Yes]], No, [[s3, a3], -4, 20, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, -4], [[t3, a3], 20, 4]], [[[a3, s3], 8, 20, Yes], [[a3, t3], 12, 20, Yes], [[s3, a3], -4, 20, No], [[t3, a3], 4, 20, Yes]], No, [[s3, a3], -4, 20, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0030","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7344501584073f301080f1e0c2f05a6d93b32b2a36bbaf7397170397f4af8642","family_id":"qf-mod44-flow-capacity-audit","seed":202611137030,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[25,1],[25,1],[25,1],[25,1]],\"edges\":[[\"a4\",\"s4\"],[\"a4\",\"t4\"],[\"s4\",\"a4\"],[\"t4\",\"a4\"]],\"flows\":[[10,1],[15,1],[30,1],[5,1]],\"sink\":\"t4\",\"source\":\"s4\",\"vertices\":[\"a4\",\"s4\",\"t4\",\"u4\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s4, a4, t4, u4], ordered directed edges=[[s4, a4], [a4, s4], [a4, t4], [t4, a4]], distinct source=s4, sink=t4, aligned capacities=[25, 25, 25, 25] and candidate flows=[30, 10, 15, 5]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 30], [[t4, a4], 25, 5]], [[[a4, s4], 10, 25, Yes], [[a4, t4], 15, 25, Yes], [[s4, a4], 30, 25, No], [[t4, a4], 5, 25, Yes]], No, [[s4, a4], 30, 25, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 30], [[t4, a4], 25, 5]], [[[a4, s4], 10, 25, Yes], [[a4, t4], 15, 25, Yes], [[s4, a4], 30, 25, No], [[t4, a4], 5, 25, Yes]], No, [[s4, a4], 30, 25, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0031","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9441d7507b1c6eb2597a16800ae5c7a1080a5072d8faadac85e58a3a7423db1a","family_id":"qf-mod44-flow-capacity-audit","seed":202611137031,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[30,1],[30,1],[30,1],[30,1]],\"edges\":[[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"s5\",\"a5\"],[\"t5\",\"a5\"]],\"flows\":[[12,1],[18,1],[24,1],[6,1]],\"sink\":\"t5\",\"source\":\"s5\",\"vertices\":[\"a5\",\"s5\",\"t5\",\"u5\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s5, a5, t5, u5], ordered directed edges=[[s5, a5], [a5, s5], [a5, t5], [t5, a5]], distinct source=s5, sink=t5, aligned capacities=[30, 30, 30, 30] and candidate flows=[24, 12, 18, 6]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], [[[a5, s5], 12, 30, Yes], [[a5, t5], 18, 30, Yes], [[s5, a5], 24, 30, Yes], [[t5, a5], 6, 30, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], [[[a5, s5], 12, 30, Yes], [[a5, t5], 18, 30, Yes], [[s5, a5], 24, 30, Yes], [[t5, a5], 6, 30, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0032","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bea2e2d2d14afeaa7b6a614cfbb2297e5193b64b4dc0ba457e0d02a5a40bfe93","family_id":"qf-mod44-flow-capacity-audit","seed":202611137032,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[35,1],[35,1],[35,1],[35,1]],\"edges\":[[\"a6\",\"s6\"],[\"a6\",\"t6\"],[\"s6\",\"a6\"],[\"t6\",\"a6\"]],\"flows\":[[14,1],[21,1],[-7,1],[7,1]],\"sink\":\"t6\",\"source\":\"s6\",\"vertices\":[\"a6\",\"s6\",\"t6\",\"u6\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s6, a6, t6, u6], ordered directed edges=[[s6, a6], [a6, s6], [a6, t6], [t6, a6]], distinct source=s6, sink=t6, aligned capacities=[35, 35, 35, 35] and candidate flows=[-7, 14, 21, 7]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, -7], [[t6, a6], 35, 7]], [[[a6, s6], 14, 35, Yes], [[a6, t6], 21, 35, Yes], [[s6, a6], -7, 35, No], [[t6, a6], 7, 35, Yes]], No, [[s6, a6], -7, 35, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, -7], [[t6, a6], 35, 7]], [[[a6, s6], 14, 35, Yes], [[a6, t6], 21, 35, Yes], [[s6, a6], -7, 35, No], [[t6, a6], 7, 35, Yes]], No, [[s6, a6], -7, 35, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0033","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87de350744c186a64063db6a2e52fd6bc97d1564fb25edbe1400768a6cde403c","family_id":"qf-mod44-flow-capacity-audit","seed":202611137033,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[40,1],[40,1],[40,1],[40,1]],\"edges\":[[\"a7\",\"s7\"],[\"a7\",\"t7\"],[\"s7\",\"a7\"],[\"t7\",\"a7\"]],\"flows\":[[16,1],[24,1],[48,1],[8,1]],\"sink\":\"t7\",\"source\":\"s7\",\"vertices\":[\"a7\",\"s7\",\"t7\",\"u7\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s7, a7, t7, u7], ordered directed edges=[[s7, a7], [a7, s7], [a7, t7], [t7, a7]], distinct source=s7, sink=t7, aligned capacities=[40, 40, 40, 40] and candidate flows=[48, 16, 24, 8]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 24], [[s7, a7], 40, 48], [[t7, a7], 40, 8]], [[[a7, s7], 16, 40, Yes], [[a7, t7], 24, 40, Yes], [[s7, a7], 48, 40, No], [[t7, a7], 8, 40, Yes]], No, [[s7, a7], 48, 40, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 24], [[s7, a7], 40, 48], [[t7, a7], 40, 8]], [[[a7, s7], 16, 40, Yes], [[a7, t7], 24, 40, Yes], [[s7, a7], 48, 40, No], [[t7, a7], 8, 40, Yes]], No, [[s7, a7], 48, 40, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0034","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bc0635cca7634ffeafbd916f1be17c12207f6d6b7e122207130f2ffdbe7eeac","family_id":"qf-mod44-flow-capacity-audit","seed":202611137034,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[45,1],[45,1],[45,1],[45,1]],\"edges\":[[\"a8\",\"s8\"],[\"a8\",\"t8\"],[\"s8\",\"a8\"],[\"t8\",\"a8\"]],\"flows\":[[18,1],[27,1],[36,1],[9,1]],\"sink\":\"t8\",\"source\":\"s8\",\"vertices\":[\"a8\",\"s8\",\"t8\",\"u8\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s8, a8, t8, u8], ordered directed edges=[[s8, a8], [a8, s8], [a8, t8], [t8, a8]], distinct source=s8, sink=t8, aligned capacities=[45, 45, 45, 45] and candidate flows=[36, 18, 27, 9]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], [[[a8, s8], 18, 45, Yes], [[a8, t8], 27, 45, Yes], [[s8, a8], 36, 45, Yes], [[t8, a8], 9, 45, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], [[[a8, s8], 18, 45, Yes], [[a8, t8], 27, 45, Yes], [[s8, a8], 36, 45, Yes], [[t8, a8], 9, 45, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0035","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"885603b559f55f2090d5c7a3b8d51afaf9c7712cf705905c3a5eba5fba41d1fa","family_id":"qf-mod44-flow-capacity-audit","seed":202611137035,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[50,1],[50,1],[50,1],[50,1]],\"edges\":[[\"a9\",\"s9\"],[\"a9\",\"t9\"],[\"s9\",\"a9\"],[\"t9\",\"a9\"]],\"flows\":[[20,1],[30,1],[-10,1],[10,1]],\"sink\":\"t9\",\"source\":\"s9\",\"vertices\":[\"a9\",\"s9\",\"t9\",\"u9\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s9, a9, t9, u9], ordered directed edges=[[s9, a9], [a9, s9], [a9, t9], [t9, a9]], distinct source=s9, sink=t9, aligned capacities=[50, 50, 50, 50] and candidate flows=[-10, 20, 30, 10]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, -10], [[t9, a9], 50, 10]], [[[a9, s9], 20, 50, Yes], [[a9, t9], 30, 50, Yes], [[s9, a9], -10, 50, No], [[t9, a9], 10, 50, Yes]], No, [[s9, a9], -10, 50, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, -10], [[t9, a9], 50, 10]], [[[a9, s9], 20, 50, Yes], [[a9, t9], 30, 50, Yes], [[s9, a9], -10, 50, No], [[t9, a9], 10, 50, Yes]], No, [[s9, a9], -10, 50, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0036","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f59df0111709405d802b293f81c4d4e4a83bbf6c72cbd214f0a2b6bd0281430b","family_id":"qf-mod44-flow-capacity-audit","seed":202611137036,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[55,1],[55,1],[55,1],[55,1]],\"edges\":[[\"a10\",\"s10\"],[\"a10\",\"t10\"],[\"s10\",\"a10\"],[\"t10\",\"a10\"]],\"flows\":[[22,1],[33,1],[66,1],[11,1]],\"sink\":\"t10\",\"source\":\"s10\",\"vertices\":[\"a10\",\"s10\",\"t10\",\"u10\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s10, a10, t10, u10], ordered directed edges=[[s10, a10], [a10, s10], [a10, t10], [t10, a10]], distinct source=s10, sink=t10, aligned capacities=[55, 55, 55, 55] and candidate flows=[66, 22, 33, 11]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 66], [[t10, a10], 55, 11]], [[[a10, s10], 22, 55, Yes], [[a10, t10], 33, 55, Yes], [[s10, a10], 66, 55, No], [[t10, a10], 11, 55, Yes]], No, [[s10, a10], 66, 55, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 66], [[t10, a10], 55, 11]], [[[a10, s10], 22, 55, Yes], [[a10, t10], 33, 55, Yes], [[s10, a10], 66, 55, No], [[t10, a10], 11, 55, Yes]], No, [[s10, a10], 66, 55, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0037","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2efb47e2aa856b27dca8d88ce290538bcdcdb284f335aea6c77e9191c06962fb","family_id":"qf-mod44-flow-capacity-audit","seed":202611137037,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[60,1],[60,1],[60,1],[60,1]],\"edges\":[[\"a11\",\"s11\"],[\"a11\",\"t11\"],[\"s11\",\"a11\"],[\"t11\",\"a11\"]],\"flows\":[[24,1],[36,1],[48,1],[12,1]],\"sink\":\"t11\",\"source\":\"s11\",\"vertices\":[\"a11\",\"s11\",\"t11\",\"u11\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s11, a11, t11, u11], ordered directed edges=[[s11, a11], [a11, s11], [a11, t11], [t11, a11]], distinct source=s11, sink=t11, aligned capacities=[60, 60, 60, 60] and candidate flows=[48, 24, 36, 12]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 36], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], [[[a11, s11], 24, 60, Yes], [[a11, t11], 36, 60, Yes], [[s11, a11], 48, 60, Yes], [[t11, a11], 12, 60, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 36], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], [[[a11, s11], 24, 60, Yes], [[a11, t11], 36, 60, Yes], [[s11, a11], 48, 60, Yes], [[t11, a11], 12, 60, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0038","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"115ecf15680002d7623a1cd3dccc4ac8e3fde0d575c27cddaefb6fc21242ece2","family_id":"qf-mod44-flow-capacity-audit","seed":202611137038,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[65,1],[65,1],[65,1],[65,1]],\"edges\":[[\"a12\",\"s12\"],[\"a12\",\"t12\"],[\"s12\",\"a12\"],[\"t12\",\"a12\"]],\"flows\":[[26,1],[39,1],[-13,1],[13,1]],\"sink\":\"t12\",\"source\":\"s12\",\"vertices\":[\"a12\",\"s12\",\"t12\",\"u12\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s12, a12, t12, u12], ordered directed edges=[[s12, a12], [a12, s12], [a12, t12], [t12, a12]], distinct source=s12, sink=t12, aligned capacities=[65, 65, 65, 65] and candidate flows=[-13, 26, 39, 13]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, -13], [[t12, a12], 65, 13]], [[[a12, s12], 26, 65, Yes], [[a12, t12], 39, 65, Yes], [[s12, a12], -13, 65, No], [[t12, a12], 13, 65, Yes]], No, [[s12, a12], -13, 65, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, -13], [[t12, a12], 65, 13]], [[[a12, s12], 26, 65, Yes], [[a12, t12], 39, 65, Yes], [[s12, a12], -13, 65, No], [[t12, a12], 13, 65, Yes]], No, [[s12, a12], -13, 65, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0039","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2990a246559f4b998862c94b681050984cd6ba96478cd7a1f60438384b0629b7","family_id":"qf-mod44-flow-capacity-audit","seed":202611137039,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[70,1],[70,1],[70,1],[70,1]],\"edges\":[[\"a13\",\"s13\"],[\"a13\",\"t13\"],[\"s13\",\"a13\"],[\"t13\",\"a13\"]],\"flows\":[[28,1],[42,1],[84,1],[14,1]],\"sink\":\"t13\",\"source\":\"s13\",\"vertices\":[\"a13\",\"s13\",\"t13\",\"u13\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s13, a13, t13, u13], ordered directed edges=[[s13, a13], [a13, s13], [a13, t13], [t13, a13]], distinct source=s13, sink=t13, aligned capacities=[70, 70, 70, 70] and candidate flows=[84, 28, 42, 14]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 42], [[s13, a13], 70, 84], [[t13, a13], 70, 14]], [[[a13, s13], 28, 70, Yes], [[a13, t13], 42, 70, Yes], [[s13, a13], 84, 70, No], [[t13, a13], 14, 70, Yes]], No, [[s13, a13], 84, 70, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 42], [[s13, a13], 70, 84], [[t13, a13], 70, 14]], [[[a13, s13], 28, 70, Yes], [[a13, t13], 42, 70, Yes], [[s13, a13], 84, 70, No], [[t13, a13], 14, 70, Yes]], No, [[s13, a13], 84, 70, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0040","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1967a64363370f618ad18e56b19488e6495858145ec7f16b0625ed954fc70f5f","family_id":"qf-mod44-flow-capacity-audit","seed":202611137040,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[75,1],[75,1],[75,1],[75,1]],\"edges\":[[\"a14\",\"s14\"],[\"a14\",\"t14\"],[\"s14\",\"a14\"],[\"t14\",\"a14\"]],\"flows\":[[30,1],[45,1],[60,1],[15,1]],\"sink\":\"t14\",\"source\":\"s14\",\"vertices\":[\"a14\",\"s14\",\"t14\",\"u14\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s14, a14, t14, u14], ordered directed edges=[[s14, a14], [a14, s14], [a14, t14], [t14, a14]], distinct source=s14, sink=t14, aligned capacities=[75, 75, 75, 75] and candidate flows=[60, 30, 45, 15]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], [[[a14, s14], 30, 75, Yes], [[a14, t14], 45, 75, Yes], [[s14, a14], 60, 75, Yes], [[t14, a14], 15, 75, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], [[[a14, s14], 30, 75, Yes], [[a14, t14], 45, 75, Yes], [[s14, a14], 60, 75, Yes], [[t14, a14], 15, 75, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0041","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7856df14468f6bf723e6e9bfba17775ec519b6e90578246823df15b23b20195","family_id":"qf-mod44-flow-capacity-audit","seed":202611137041,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[80,1],[80,1],[80,1],[80,1]],\"edges\":[[\"a15\",\"s15\"],[\"a15\",\"t15\"],[\"s15\",\"a15\"],[\"t15\",\"a15\"]],\"flows\":[[32,1],[48,1],[-16,1],[16,1]],\"sink\":\"t15\",\"source\":\"s15\",\"vertices\":[\"a15\",\"s15\",\"t15\",\"u15\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s15, a15, t15, u15], ordered directed edges=[[s15, a15], [a15, s15], [a15, t15], [t15, a15]], distinct source=s15, sink=t15, aligned capacities=[80, 80, 80, 80] and candidate flows=[-16, 32, 48, 16]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, -16], [[t15, a15], 80, 16]], [[[a15, s15], 32, 80, Yes], [[a15, t15], 48, 80, Yes], [[s15, a15], -16, 80, No], [[t15, a15], 16, 80, Yes]], No, [[s15, a15], -16, 80, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, -16], [[t15, a15], 80, 16]], [[[a15, s15], 32, 80, Yes], [[a15, t15], 48, 80, Yes], [[s15, a15], -16, 80, No], [[t15, a15], 16, 80, Yes]], No, [[s15, a15], -16, 80, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0042","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ae10d2bc097bd90dc558eddec62926ce4853754325f27232cc5032155ff6650","family_id":"qf-mod44-flow-capacity-audit","seed":202611137042,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[85,1],[85,1],[85,1],[85,1]],\"edges\":[[\"a16\",\"s16\"],[\"a16\",\"t16\"],[\"s16\",\"a16\"],[\"t16\",\"a16\"]],\"flows\":[[34,1],[51,1],[102,1],[17,1]],\"sink\":\"t16\",\"source\":\"s16\",\"vertices\":[\"a16\",\"s16\",\"t16\",\"u16\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s16, a16, t16, u16], ordered directed edges=[[s16, a16], [a16, s16], [a16, t16], [t16, a16]], distinct source=s16, sink=t16, aligned capacities=[85, 85, 85, 85] and candidate flows=[102, 34, 51, 17]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 102], [[t16, a16], 85, 17]], [[[a16, s16], 34, 85, Yes], [[a16, t16], 51, 85, Yes], [[s16, a16], 102, 85, No], [[t16, a16], 17, 85, Yes]], No, [[s16, a16], 102, 85, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 102], [[t16, a16], 85, 17]], [[[a16, s16], 34, 85, Yes], [[a16, t16], 51, 85, Yes], [[s16, a16], 102, 85, No], [[t16, a16], 17, 85, Yes]], No, [[s16, a16], 102, 85, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0043","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a04262683ef242d46ff05a4abcd349728f1eaae0771269283ae932b5141f4d71","family_id":"qf-mod44-flow-capacity-audit","seed":202611137043,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[90,1],[90,1],[90,1],[90,1]],\"edges\":[[\"a17\",\"s17\"],[\"a17\",\"t17\"],[\"s17\",\"a17\"],[\"t17\",\"a17\"]],\"flows\":[[36,1],[54,1],[72,1],[18,1]],\"sink\":\"t17\",\"source\":\"s17\",\"vertices\":[\"a17\",\"s17\",\"t17\",\"u17\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s17, a17, t17, u17], ordered directed edges=[[s17, a17], [a17, s17], [a17, t17], [t17, a17]], distinct source=s17, sink=t17, aligned capacities=[90, 90, 90, 90] and candidate flows=[72, 36, 54, 18]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 54], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], [[[a17, s17], 36, 90, Yes], [[a17, t17], 54, 90, Yes], [[s17, a17], 72, 90, Yes], [[t17, a17], 18, 90, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 54], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], [[[a17, s17], 36, 90, Yes], [[a17, t17], 54, 90, Yes], [[s17, a17], 72, 90, Yes], [[t17, a17], 18, 90, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0044","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7ce14deb5ed9b49155016a59b5c076dc0669edd8a948363307b2a6314877238","family_id":"qf-mod44-flow-capacity-audit","seed":202611137044,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[95,1],[95,1],[95,1],[95,1]],\"edges\":[[\"a18\",\"s18\"],[\"a18\",\"t18\"],[\"s18\",\"a18\"],[\"t18\",\"a18\"]],\"flows\":[[38,1],[57,1],[-19,1],[19,1]],\"sink\":\"t18\",\"source\":\"s18\",\"vertices\":[\"a18\",\"s18\",\"t18\",\"u18\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s18, a18, t18, u18], ordered directed edges=[[s18, a18], [a18, s18], [a18, t18], [t18, a18]], distinct source=s18, sink=t18, aligned capacities=[95, 95, 95, 95] and candidate flows=[-19, 38, 57, 19]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, -19], [[t18, a18], 95, 19]], [[[a18, s18], 38, 95, Yes], [[a18, t18], 57, 95, Yes], [[s18, a18], -19, 95, No], [[t18, a18], 19, 95, Yes]], No, [[s18, a18], -19, 95, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, -19], [[t18, a18], 95, 19]], [[[a18, s18], 38, 95, Yes], [[a18, t18], 57, 95, Yes], [[s18, a18], -19, 95, No], [[t18, a18], 19, 95, Yes]], No, [[s18, a18], -19, 95, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0045","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7a5736fd59925d989ca200e4b6c0ee053e6b04f8277456d97571e370e6526c9","family_id":"qf-mod44-flow-capacity-audit","seed":202611137045,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[100,1],[100,1],[100,1],[100,1]],\"edges\":[[\"a19\",\"s19\"],[\"a19\",\"t19\"],[\"s19\",\"a19\"],[\"t19\",\"a19\"]],\"flows\":[[40,1],[60,1],[120,1],[20,1]],\"sink\":\"t19\",\"source\":\"s19\",\"vertices\":[\"a19\",\"s19\",\"t19\",\"u19\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s19, a19, t19, u19], ordered directed edges=[[s19, a19], [a19, s19], [a19, t19], [t19, a19]], distinct source=s19, sink=t19, aligned capacities=[100, 100, 100, 100] and candidate flows=[120, 40, 60, 20]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 60], [[s19, a19], 100, 120], [[t19, a19], 100, 20]], [[[a19, s19], 40, 100, Yes], [[a19, t19], 60, 100, Yes], [[s19, a19], 120, 100, No], [[t19, a19], 20, 100, Yes]], No, [[s19, a19], 120, 100, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 60], [[s19, a19], 100, 120], [[t19, a19], 100, 20]], [[[a19, s19], 40, 100, Yes], [[a19, t19], 60, 100, Yes], [[s19, a19], 120, 100, No], [[t19, a19], 20, 100, Yes]], No, [[s19, a19], 120, 100, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0046","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d4dd8a87d871bac64bf33832757fd738ae15c9d5088ee79945f402ad06caaa1","family_id":"qf-mod44-flow-capacity-audit","seed":202611137046,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[105,1],[105,1],[105,1],[105,1]],\"edges\":[[\"a20\",\"s20\"],[\"a20\",\"t20\"],[\"s20\",\"a20\"],[\"t20\",\"a20\"]],\"flows\":[[42,1],[63,1],[84,1],[21,1]],\"sink\":\"t20\",\"source\":\"s20\",\"vertices\":[\"a20\",\"s20\",\"t20\",\"u20\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s20, a20, t20, u20], ordered directed edges=[[s20, a20], [a20, s20], [a20, t20], [t20, a20]], distinct source=s20, sink=t20, aligned capacities=[105, 105, 105, 105] and candidate flows=[84, 42, 63, 21]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], [[[a20, s20], 42, 105, Yes], [[a20, t20], 63, 105, Yes], [[s20, a20], 84, 105, Yes], [[t20, a20], 21, 105, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], [[[a20, s20], 42, 105, Yes], [[a20, t20], 63, 105, Yes], [[s20, a20], 84, 105, Yes], [[t20, a20], 21, 105, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0047","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1034492af45d5136ce06bf6fe4d6ce87b5adc7bb6fd0b3f2d1af9d69413999b","family_id":"qf-mod44-flow-capacity-audit","seed":202611137047,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[110,1],[110,1],[110,1],[110,1]],\"edges\":[[\"a21\",\"s21\"],[\"a21\",\"t21\"],[\"s21\",\"a21\"],[\"t21\",\"a21\"]],\"flows\":[[44,1],[66,1],[-22,1],[22,1]],\"sink\":\"t21\",\"source\":\"s21\",\"vertices\":[\"a21\",\"s21\",\"t21\",\"u21\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s21, a21, t21, u21], ordered directed edges=[[s21, a21], [a21, s21], [a21, t21], [t21, a21]], distinct source=s21, sink=t21, aligned capacities=[110, 110, 110, 110] and candidate flows=[-22, 44, 66, 22]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 66], [[s21, a21], 110, -22], [[t21, a21], 110, 22]], [[[a21, s21], 44, 110, Yes], [[a21, t21], 66, 110, Yes], [[s21, a21], -22, 110, No], [[t21, a21], 22, 110, Yes]], No, [[s21, a21], -22, 110, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 66], [[s21, a21], 110, -22], [[t21, a21], 110, 22]], [[[a21, s21], 44, 110, Yes], [[a21, t21], 66, 110, Yes], [[s21, a21], -22, 110, No], [[t21, a21], 22, 110, Yes]], No, [[s21, a21], -22, 110, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0048","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38b9b619c1e12ab945f212629014eacb4a54bce7856994d4599de1262392048a","family_id":"qf-mod44-flow-capacity-audit","seed":202611137048,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[115,1],[115,1],[115,1],[115,1]],\"edges\":[[\"a22\",\"s22\"],[\"a22\",\"t22\"],[\"s22\",\"a22\"],[\"t22\",\"a22\"]],\"flows\":[[46,1],[69,1],[138,1],[23,1]],\"sink\":\"t22\",\"source\":\"s22\",\"vertices\":[\"a22\",\"s22\",\"t22\",\"u22\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s22, a22, t22, u22], ordered directed edges=[[s22, a22], [a22, s22], [a22, t22], [t22, a22]], distinct source=s22, sink=t22, aligned capacities=[115, 115, 115, 115] and candidate flows=[138, 46, 69, 23]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 138], [[t22, a22], 115, 23]], [[[a22, s22], 46, 115, Yes], [[a22, t22], 69, 115, Yes], [[s22, a22], 138, 115, No], [[t22, a22], 23, 115, Yes]], No, [[s22, a22], 138, 115, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 138], [[t22, a22], 115, 23]], [[[a22, s22], 46, 115, Yes], [[a22, t22], 69, 115, Yes], [[s22, a22], 138, 115, No], [[t22, a22], 23, 115, Yes]], No, [[s22, a22], 138, 115, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0049","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"068fee52962af5fd9f83e4b8acbf428848dd71ab48f2b1a1962b06851cbd0093","family_id":"qf-mod44-flow-capacity-audit","seed":202611137049,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[120,1],[120,1],[120,1],[120,1]],\"edges\":[[\"a23\",\"s23\"],[\"a23\",\"t23\"],[\"s23\",\"a23\"],[\"t23\",\"a23\"]],\"flows\":[[48,1],[72,1],[96,1],[24,1]],\"sink\":\"t23\",\"source\":\"s23\",\"vertices\":[\"a23\",\"s23\",\"t23\",\"u23\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s23, a23, t23, u23], ordered directed edges=[[s23, a23], [a23, s23], [a23, t23], [t23, a23]], distinct source=s23, sink=t23, aligned capacities=[120, 120, 120, 120] and candidate flows=[96, 48, 72, 24]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], [[[a23, s23], 48, 120, Yes], [[a23, t23], 72, 120, Yes], [[s23, a23], 96, 120, Yes], [[t23, a23], 24, 120, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], [[[a23, s23], 48, 120, Yes], [[a23, t23], 72, 120, Yes], [[s23, a23], 96, 120, Yes], [[t23, a23], 24, 120, Yes]], Yes, null, every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0050","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"882ff96f6c85cd9042a446f17816a59996920cce999f022834c43bfbc819a1fa","family_id":"qf-mod44-flow-capacity-audit","seed":202611137050,"semantic_template":"qf-mod44-flow-capacity-audit/{\"capacities\":[[125,1],[125,1],[125,1],[125,1]],\"edges\":[[\"a24\",\"s24\"],[\"a24\",\"t24\"],[\"s24\",\"a24\"],[\"t24\",\"a24\"]],\"flows\":[[50,1],[75,1],[-25,1],[25,1]],\"sink\":\"t24\",\"source\":\"s24\",\"vertices\":[\"a24\",\"s24\",\"t24\",\"u24\"]}"},"fields":{"title":"Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity","stem":"Use vertices=[s24, a24, t24, u24], ordered directed edges=[[s24, a24], [a24, s24], [a24, t24], [t24, a24]], distinct source=s24, sink=t24, aligned capacities=[125, 125, 125, 125] and candidate flows=[-25, 50, 75, 25]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate).","answer":{"kind":"expression","value":"(s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, -25], [[t24, a24], 125, 25]], [[[a24, s24], 50, 125, Yes], [[a24, t24], 75, 125, Yes], [[s24, a24], -25, 125, No], [[t24, a24], 25, 125, Yes]], No, [[s24, a24], -25, 125, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)","explanation":"Keep network rows [edge,capacity,flow] distinct from audit rows [edge,flow,capacity,pass]. Report the first failing edge in canonical order, or null. Negative/over-capacity candidates are rejected, not rewritten; local capacity success alone does not establish conservation. The exact result, retaining the requested order, is (s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, -25], [[t24, a24], 125, 25]], [[[a24, s24], 50, 125, Yes], [[a24, t24], 75, 125, Yes], [[s24, a24], -25, 125, No], [[t24, a24], 25, 125, Yes]], No, [[s24, a24], -25, 125, No], every_edge_requires_zero_less_equal_flow_less_equal_capacity)."},"skills":["Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity"],"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-20261113-r07-finite-flow-network-0051","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad0d55b626df9fd75b975acdd76b6197bfb8bc48da16ff0bd01162b9624640f6","family_id":"qf-mod44-flow-conservation-audit","seed":202611137051,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[5,1],[5,1],[5,1],[5,1]],\"edges\":[[\"a0\",\"s0\"],[\"a0\",\"t0\"],[\"s0\",\"a0\"],[\"t0\",\"a0\"]],\"flows\":[[2,1],[3,1],[4,1],[1,1]],\"sink\":\"t0\",\"source\":\"s0\",\"vertices\":[\"a0\",\"s0\",\"t0\",\"u0\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s0, a0, t0, u0], ordered directed edges=[[s0, a0], [a0, s0], [a0, t0], [t0, a0]], distinct source=s0, sink=t0, aligned capacities=[5, 5, 5, 5] and candidate flows=[4, 2, 3, 1]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, 4], [[t0, a0], 5, 1]], [[a0, 5, 5, Yes], [u0, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, 4], [[t0, a0], 5, 1]], [[a0, 5, 5, Yes], [u0, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0052","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d58a405404b3f52b4ca1768b65da346953e8865701e0514a8c9e07dd4c0f8cc1","family_id":"qf-mod44-flow-conservation-audit","seed":202611137052,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[10,1],[10,1],[10,1],[10,1]],\"edges\":[[\"a1\",\"s1\"],[\"a1\",\"t1\"],[\"s1\",\"a1\"],[\"t1\",\"a1\"]],\"flows\":[[4,1],[4,1],[8,1],[2,1]],\"sink\":\"t1\",\"source\":\"s1\",\"vertices\":[\"a1\",\"s1\",\"t1\",\"u1\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s1, a1, t1, u1], ordered directed edges=[[s1, a1], [a1, s1], [a1, t1], [t1, a1]], distinct source=s1, sink=t1, aligned capacities=[10, 10, 10, 10] and candidate flows=[8, 4, 4, 2]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 4], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], [[a1, 10, 8, No], [u1, 0, 0, Yes]], No, [a1, 10, 8, No], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 4], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], [[a1, 10, 8, No], [u1, 0, 0, Yes]], No, [a1, 10, 8, No], every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0053","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cca58fa2bff1e3d824dcee5b7577f48a4426161abe4d8e6c453fd639ad67ad59","family_id":"qf-mod44-flow-conservation-audit","seed":202611137053,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[15,1],[15,1],[15,1],[15,1]],\"edges\":[[\"a2\",\"s2\"],[\"a2\",\"t2\"],[\"s2\",\"a2\"],[\"t2\",\"a2\"]],\"flows\":[[6,1],[9,1],[12,1],[3,1]],\"sink\":\"t2\",\"source\":\"s2\",\"vertices\":[\"a2\",\"s2\",\"t2\",\"u2\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s2, a2, t2, u2], ordered directed edges=[[s2, a2], [a2, s2], [a2, t2], [t2, a2]], distinct source=s2, sink=t2, aligned capacities=[15, 15, 15, 15] and candidate flows=[12, 6, 9, 3]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], [[a2, 15, 15, Yes], [u2, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], [[a2, 15, 15, Yes], [u2, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0054","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62e605f13f8795fb236c02ab28b022b66e6d63ed2bb84decea3fde729821487e","family_id":"qf-mod44-flow-conservation-audit","seed":202611137054,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[20,1],[20,1],[20,1],[20,1]],\"edges\":[[\"a3\",\"s3\"],[\"a3\",\"t3\"],[\"s3\",\"a3\"],[\"t3\",\"a3\"]],\"flows\":[[8,1],[8,1],[16,1],[4,1]],\"sink\":\"t3\",\"source\":\"s3\",\"vertices\":[\"a3\",\"s3\",\"t3\",\"u3\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s3, a3, t3, u3], ordered directed edges=[[s3, a3], [a3, s3], [a3, t3], [t3, a3]], distinct source=s3, sink=t3, aligned capacities=[20, 20, 20, 20] and candidate flows=[16, 8, 8, 4]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 8], [[s3, a3], 20, 16], [[t3, a3], 20, 4]], [[a3, 20, 16, No], [u3, 0, 0, Yes]], No, [a3, 20, 16, No], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 8], [[s3, a3], 20, 16], [[t3, a3], 20, 4]], [[a3, 20, 16, No], [u3, 0, 0, Yes]], No, [a3, 20, 16, No], every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0055","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22987b7a2acde21ac85c856b55ac71375c1c1d4522aaacb335e305171274d8fa","family_id":"qf-mod44-flow-conservation-audit","seed":202611137055,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[25,1],[25,1],[25,1],[25,1]],\"edges\":[[\"a4\",\"s4\"],[\"a4\",\"t4\"],[\"s4\",\"a4\"],[\"t4\",\"a4\"]],\"flows\":[[10,1],[15,1],[20,1],[5,1]],\"sink\":\"t4\",\"source\":\"s4\",\"vertices\":[\"a4\",\"s4\",\"t4\",\"u4\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s4, a4, t4, u4], ordered directed edges=[[s4, a4], [a4, s4], [a4, t4], [t4, a4]], distinct source=s4, sink=t4, aligned capacities=[25, 25, 25, 25] and candidate flows=[20, 10, 15, 5]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 20], [[t4, a4], 25, 5]], [[a4, 25, 25, Yes], [u4, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 20], [[t4, a4], 25, 5]], [[a4, 25, 25, Yes], [u4, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0056","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58c8fd8b39b03b85c271628b862d4f470ad0fef0b8307c94bff818cb3bd8d6b2","family_id":"qf-mod44-flow-conservation-audit","seed":202611137056,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[30,1],[30,1],[30,1],[30,1]],\"edges\":[[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"s5\",\"a5\"],[\"t5\",\"a5\"]],\"flows\":[[12,1],[12,1],[24,1],[6,1]],\"sink\":\"t5\",\"source\":\"s5\",\"vertices\":[\"a5\",\"s5\",\"t5\",\"u5\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s5, a5, t5, u5], ordered directed edges=[[s5, a5], [a5, s5], [a5, t5], [t5, a5]], distinct source=s5, sink=t5, aligned capacities=[30, 30, 30, 30] and candidate flows=[24, 12, 12, 6]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 12], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], [[a5, 30, 24, No], [u5, 0, 0, Yes]], No, [a5, 30, 24, No], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 12], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], [[a5, 30, 24, No], [u5, 0, 0, Yes]], No, [a5, 30, 24, No], every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0057","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce051f6ad314e917d9afd973986c8db144b52415ef5fbbd29c5c34c9fcfa0271","family_id":"qf-mod44-flow-conservation-audit","seed":202611137057,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[35,1],[35,1],[35,1],[35,1]],\"edges\":[[\"a6\",\"s6\"],[\"a6\",\"t6\"],[\"s6\",\"a6\"],[\"t6\",\"a6\"]],\"flows\":[[14,1],[21,1],[28,1],[7,1]],\"sink\":\"t6\",\"source\":\"s6\",\"vertices\":[\"a6\",\"s6\",\"t6\",\"u6\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s6, a6, t6, u6], ordered directed edges=[[s6, a6], [a6, s6], [a6, t6], [t6, a6]], distinct source=s6, sink=t6, aligned capacities=[35, 35, 35, 35] and candidate flows=[28, 14, 21, 7]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], [[a6, 35, 35, Yes], [u6, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], [[a6, 35, 35, Yes], [u6, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0058","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92ec748c311705ab272f92781b5eb3749602d3f371df03318cad29ffbae404bd","family_id":"qf-mod44-flow-conservation-audit","seed":202611137058,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[40,1],[40,1],[40,1],[40,1]],\"edges\":[[\"a7\",\"s7\"],[\"a7\",\"t7\"],[\"s7\",\"a7\"],[\"t7\",\"a7\"]],\"flows\":[[16,1],[16,1],[32,1],[8,1]],\"sink\":\"t7\",\"source\":\"s7\",\"vertices\":[\"a7\",\"s7\",\"t7\",\"u7\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s7, a7, t7, u7], ordered directed edges=[[s7, a7], [a7, s7], [a7, t7], [t7, a7]], distinct source=s7, sink=t7, aligned capacities=[40, 40, 40, 40] and candidate flows=[32, 16, 16, 8]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 16], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], [[a7, 40, 32, No], [u7, 0, 0, Yes]], No, [a7, 40, 32, No], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 16], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], [[a7, 40, 32, No], [u7, 0, 0, Yes]], No, [a7, 40, 32, No], every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0059","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7889b7ff11778ab4177828cc0d84b0e26fb6922c938d3714ce36da7caef35034","family_id":"qf-mod44-flow-conservation-audit","seed":202611137059,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[45,1],[45,1],[45,1],[45,1]],\"edges\":[[\"a8\",\"s8\"],[\"a8\",\"t8\"],[\"s8\",\"a8\"],[\"t8\",\"a8\"]],\"flows\":[[18,1],[27,1],[36,1],[9,1]],\"sink\":\"t8\",\"source\":\"s8\",\"vertices\":[\"a8\",\"s8\",\"t8\",\"u8\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s8, a8, t8, u8], ordered directed edges=[[s8, a8], [a8, s8], [a8, t8], [t8, a8]], distinct source=s8, sink=t8, aligned capacities=[45, 45, 45, 45] and candidate flows=[36, 18, 27, 9]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], [[a8, 45, 45, Yes], [u8, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], [[a8, 45, 45, Yes], [u8, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0060","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60b64b68ce517902a99c3137d9d82635feea669fccb8292c8aee01ff162738d9","family_id":"qf-mod44-flow-conservation-audit","seed":202611137060,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[50,1],[50,1],[50,1],[50,1]],\"edges\":[[\"a9\",\"s9\"],[\"a9\",\"t9\"],[\"s9\",\"a9\"],[\"t9\",\"a9\"]],\"flows\":[[20,1],[20,1],[40,1],[10,1]],\"sink\":\"t9\",\"source\":\"s9\",\"vertices\":[\"a9\",\"s9\",\"t9\",\"u9\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s9, a9, t9, u9], ordered directed edges=[[s9, a9], [a9, s9], [a9, t9], [t9, a9]], distinct source=s9, sink=t9, aligned capacities=[50, 50, 50, 50] and candidate flows=[40, 20, 20, 10]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 20], [[s9, a9], 50, 40], [[t9, a9], 50, 10]], [[a9, 50, 40, No], [u9, 0, 0, Yes]], No, [a9, 50, 40, No], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 20], [[s9, a9], 50, 40], [[t9, a9], 50, 10]], [[a9, 50, 40, No], [u9, 0, 0, Yes]], No, [a9, 50, 40, No], every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0061","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1f6aea700d11de5ac7929b032a57974d4d84e1912e2d7532f60a5b74e021a5d","family_id":"qf-mod44-flow-conservation-audit","seed":202611137061,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[55,1],[55,1],[55,1],[55,1]],\"edges\":[[\"a10\",\"s10\"],[\"a10\",\"t10\"],[\"s10\",\"a10\"],[\"t10\",\"a10\"]],\"flows\":[[22,1],[33,1],[44,1],[11,1]],\"sink\":\"t10\",\"source\":\"s10\",\"vertices\":[\"a10\",\"s10\",\"t10\",\"u10\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s10, a10, t10, u10], ordered directed edges=[[s10, a10], [a10, s10], [a10, t10], [t10, a10]], distinct source=s10, sink=t10, aligned capacities=[55, 55, 55, 55] and candidate flows=[44, 22, 33, 11]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 44], [[t10, a10], 55, 11]], [[a10, 55, 55, Yes], [u10, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 44], [[t10, a10], 55, 11]], [[a10, 55, 55, Yes], [u10, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0062","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4886f397cf637b8556b21989f6fd38c5355d4afa9e694c467d50feaea095f2b9","family_id":"qf-mod44-flow-conservation-audit","seed":202611137062,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[60,1],[60,1],[60,1],[60,1]],\"edges\":[[\"a11\",\"s11\"],[\"a11\",\"t11\"],[\"s11\",\"a11\"],[\"t11\",\"a11\"]],\"flows\":[[24,1],[24,1],[48,1],[12,1]],\"sink\":\"t11\",\"source\":\"s11\",\"vertices\":[\"a11\",\"s11\",\"t11\",\"u11\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s11, a11, t11, u11], ordered directed edges=[[s11, a11], [a11, s11], [a11, t11], [t11, a11]], distinct source=s11, sink=t11, aligned capacities=[60, 60, 60, 60] and candidate flows=[48, 24, 24, 12]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 24], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], [[a11, 60, 48, No], [u11, 0, 0, Yes]], No, [a11, 60, 48, No], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 24], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], [[a11, 60, 48, No], [u11, 0, 0, Yes]], No, [a11, 60, 48, No], every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0063","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2fd4ec63a71a84ddd162d9c79e82e43d2363931bd15afada67192117582513a","family_id":"qf-mod44-flow-conservation-audit","seed":202611137063,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[65,1],[65,1],[65,1],[65,1]],\"edges\":[[\"a12\",\"s12\"],[\"a12\",\"t12\"],[\"s12\",\"a12\"],[\"t12\",\"a12\"]],\"flows\":[[26,1],[39,1],[52,1],[13,1]],\"sink\":\"t12\",\"source\":\"s12\",\"vertices\":[\"a12\",\"s12\",\"t12\",\"u12\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s12, a12, t12, u12], ordered directed edges=[[s12, a12], [a12, s12], [a12, t12], [t12, a12]], distinct source=s12, sink=t12, aligned capacities=[65, 65, 65, 65] and candidate flows=[52, 26, 39, 13]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, 52], [[t12, a12], 65, 13]], [[a12, 65, 65, Yes], [u12, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, 52], [[t12, a12], 65, 13]], [[a12, 65, 65, Yes], [u12, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0064","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53087f600a42cfb663706ecb2c0e4fcfa215fad8cf27a057b2b98f544c1c825e","family_id":"qf-mod44-flow-conservation-audit","seed":202611137064,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[70,1],[70,1],[70,1],[70,1]],\"edges\":[[\"a13\",\"s13\"],[\"a13\",\"t13\"],[\"s13\",\"a13\"],[\"t13\",\"a13\"]],\"flows\":[[28,1],[28,1],[56,1],[14,1]],\"sink\":\"t13\",\"source\":\"s13\",\"vertices\":[\"a13\",\"s13\",\"t13\",\"u13\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s13, a13, t13, u13], ordered directed edges=[[s13, a13], [a13, s13], [a13, t13], [t13, a13]], distinct source=s13, sink=t13, aligned capacities=[70, 70, 70, 70] and candidate flows=[56, 28, 28, 14]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 28], [[s13, a13], 70, 56], [[t13, a13], 70, 14]], [[a13, 70, 56, No], [u13, 0, 0, Yes]], No, [a13, 70, 56, No], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 28], [[s13, a13], 70, 56], [[t13, a13], 70, 14]], [[a13, 70, 56, No], [u13, 0, 0, Yes]], No, [a13, 70, 56, No], every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0065","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e26e3237507ca20dd448aafff91992fc593d97e0abc3124ba28f555e0333ce84","family_id":"qf-mod44-flow-conservation-audit","seed":202611137065,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[75,1],[75,1],[75,1],[75,1]],\"edges\":[[\"a14\",\"s14\"],[\"a14\",\"t14\"],[\"s14\",\"a14\"],[\"t14\",\"a14\"]],\"flows\":[[30,1],[45,1],[60,1],[15,1]],\"sink\":\"t14\",\"source\":\"s14\",\"vertices\":[\"a14\",\"s14\",\"t14\",\"u14\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s14, a14, t14, u14], ordered directed edges=[[s14, a14], [a14, s14], [a14, t14], [t14, a14]], distinct source=s14, sink=t14, aligned capacities=[75, 75, 75, 75] and candidate flows=[60, 30, 45, 15]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], [[a14, 75, 75, Yes], [u14, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], [[a14, 75, 75, Yes], [u14, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0066","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e5fa6188deb1ab1bf6b8c3e9daf49b10bb08ee6725810769dcb3ca3ef491c38","family_id":"qf-mod44-flow-conservation-audit","seed":202611137066,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[80,1],[80,1],[80,1],[80,1]],\"edges\":[[\"a15\",\"s15\"],[\"a15\",\"t15\"],[\"s15\",\"a15\"],[\"t15\",\"a15\"]],\"flows\":[[32,1],[32,1],[64,1],[16,1]],\"sink\":\"t15\",\"source\":\"s15\",\"vertices\":[\"a15\",\"s15\",\"t15\",\"u15\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s15, a15, t15, u15], ordered directed edges=[[s15, a15], [a15, s15], [a15, t15], [t15, a15]], distinct source=s15, sink=t15, aligned capacities=[80, 80, 80, 80] and candidate flows=[64, 32, 32, 16]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 32], [[s15, a15], 80, 64], [[t15, a15], 80, 16]], [[a15, 80, 64, No], [u15, 0, 0, Yes]], No, [a15, 80, 64, No], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 32], [[s15, a15], 80, 64], [[t15, a15], 80, 16]], [[a15, 80, 64, No], [u15, 0, 0, Yes]], No, [a15, 80, 64, No], every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0067","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5d403d88885e1dc8157c059c2996d8e913ceebef7b7cca20dc50ba0cba1520b","family_id":"qf-mod44-flow-conservation-audit","seed":202611137067,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[85,1],[85,1],[85,1],[85,1]],\"edges\":[[\"a16\",\"s16\"],[\"a16\",\"t16\"],[\"s16\",\"a16\"],[\"t16\",\"a16\"]],\"flows\":[[34,1],[51,1],[68,1],[17,1]],\"sink\":\"t16\",\"source\":\"s16\",\"vertices\":[\"a16\",\"s16\",\"t16\",\"u16\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s16, a16, t16, u16], ordered directed edges=[[s16, a16], [a16, s16], [a16, t16], [t16, a16]], distinct source=s16, sink=t16, aligned capacities=[85, 85, 85, 85] and candidate flows=[68, 34, 51, 17]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 68], [[t16, a16], 85, 17]], [[a16, 85, 85, Yes], [u16, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 68], [[t16, a16], 85, 17]], [[a16, 85, 85, Yes], [u16, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0068","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b73a3003e96cd06db0a28a3c7c2e255caec054e1b082f03e3a59be394cbd466f","family_id":"qf-mod44-flow-conservation-audit","seed":202611137068,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[90,1],[90,1],[90,1],[90,1]],\"edges\":[[\"a17\",\"s17\"],[\"a17\",\"t17\"],[\"s17\",\"a17\"],[\"t17\",\"a17\"]],\"flows\":[[36,1],[36,1],[72,1],[18,1]],\"sink\":\"t17\",\"source\":\"s17\",\"vertices\":[\"a17\",\"s17\",\"t17\",\"u17\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s17, a17, t17, u17], ordered directed edges=[[s17, a17], [a17, s17], [a17, t17], [t17, a17]], distinct source=s17, sink=t17, aligned capacities=[90, 90, 90, 90] and candidate flows=[72, 36, 36, 18]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 36], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], [[a17, 90, 72, No], [u17, 0, 0, Yes]], No, [a17, 90, 72, No], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 36], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], [[a17, 90, 72, No], [u17, 0, 0, Yes]], No, [a17, 90, 72, No], every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0069","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cda0e12b7fadf4c0cd1b0b15feee70e523eb9e73ded948fc569b351473f88926","family_id":"qf-mod44-flow-conservation-audit","seed":202611137069,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[95,1],[95,1],[95,1],[95,1]],\"edges\":[[\"a18\",\"s18\"],[\"a18\",\"t18\"],[\"s18\",\"a18\"],[\"t18\",\"a18\"]],\"flows\":[[38,1],[57,1],[76,1],[19,1]],\"sink\":\"t18\",\"source\":\"s18\",\"vertices\":[\"a18\",\"s18\",\"t18\",\"u18\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s18, a18, t18, u18], ordered directed edges=[[s18, a18], [a18, s18], [a18, t18], [t18, a18]], distinct source=s18, sink=t18, aligned capacities=[95, 95, 95, 95] and candidate flows=[76, 38, 57, 19]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, 76], [[t18, a18], 95, 19]], [[a18, 95, 95, Yes], [u18, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, 76], [[t18, a18], 95, 19]], [[a18, 95, 95, Yes], [u18, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0070","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56dd12fd691baaa0fa1b07bb9c6421d529b4f9de6ea92792557a5b1661c77986","family_id":"qf-mod44-flow-conservation-audit","seed":202611137070,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[100,1],[100,1],[100,1],[100,1]],\"edges\":[[\"a19\",\"s19\"],[\"a19\",\"t19\"],[\"s19\",\"a19\"],[\"t19\",\"a19\"]],\"flows\":[[40,1],[40,1],[80,1],[20,1]],\"sink\":\"t19\",\"source\":\"s19\",\"vertices\":[\"a19\",\"s19\",\"t19\",\"u19\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s19, a19, t19, u19], ordered directed edges=[[s19, a19], [a19, s19], [a19, t19], [t19, a19]], distinct source=s19, sink=t19, aligned capacities=[100, 100, 100, 100] and candidate flows=[80, 40, 40, 20]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 40], [[s19, a19], 100, 80], [[t19, a19], 100, 20]], [[a19, 100, 80, No], [u19, 0, 0, Yes]], No, [a19, 100, 80, No], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 40], [[s19, a19], 100, 80], [[t19, a19], 100, 20]], [[a19, 100, 80, No], [u19, 0, 0, Yes]], No, [a19, 100, 80, No], every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0071","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55958c54c6ddc2dace5753676d56bedaf9667be64daff8e6a24e99a7846093c9","family_id":"qf-mod44-flow-conservation-audit","seed":202611137071,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[105,1],[105,1],[105,1],[105,1]],\"edges\":[[\"a20\",\"s20\"],[\"a20\",\"t20\"],[\"s20\",\"a20\"],[\"t20\",\"a20\"]],\"flows\":[[42,1],[63,1],[84,1],[21,1]],\"sink\":\"t20\",\"source\":\"s20\",\"vertices\":[\"a20\",\"s20\",\"t20\",\"u20\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s20, a20, t20, u20], ordered directed edges=[[s20, a20], [a20, s20], [a20, t20], [t20, a20]], distinct source=s20, sink=t20, aligned capacities=[105, 105, 105, 105] and candidate flows=[84, 42, 63, 21]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], [[a20, 105, 105, Yes], [u20, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], [[a20, 105, 105, Yes], [u20, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0072","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"731639e3c30bd36a983f67ce8f4f03a5a4db11dd7429541156a5feaf73617ea9","family_id":"qf-mod44-flow-conservation-audit","seed":202611137072,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[110,1],[110,1],[110,1],[110,1]],\"edges\":[[\"a21\",\"s21\"],[\"a21\",\"t21\"],[\"s21\",\"a21\"],[\"t21\",\"a21\"]],\"flows\":[[44,1],[44,1],[88,1],[22,1]],\"sink\":\"t21\",\"source\":\"s21\",\"vertices\":[\"a21\",\"s21\",\"t21\",\"u21\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s21, a21, t21, u21], ordered directed edges=[[s21, a21], [a21, s21], [a21, t21], [t21, a21]], distinct source=s21, sink=t21, aligned capacities=[110, 110, 110, 110] and candidate flows=[88, 44, 44, 22]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 44], [[s21, a21], 110, 88], [[t21, a21], 110, 22]], [[a21, 110, 88, No], [u21, 0, 0, Yes]], No, [a21, 110, 88, No], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 44], [[s21, a21], 110, 88], [[t21, a21], 110, 22]], [[a21, 110, 88, No], [u21, 0, 0, Yes]], No, [a21, 110, 88, No], every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0073","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd1ee89153dcf01002b5a553925e1eb4f149ee28c62c23142bd8cd1938ea82cb","family_id":"qf-mod44-flow-conservation-audit","seed":202611137073,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[115,1],[115,1],[115,1],[115,1]],\"edges\":[[\"a22\",\"s22\"],[\"a22\",\"t22\"],[\"s22\",\"a22\"],[\"t22\",\"a22\"]],\"flows\":[[46,1],[69,1],[92,1],[23,1]],\"sink\":\"t22\",\"source\":\"s22\",\"vertices\":[\"a22\",\"s22\",\"t22\",\"u22\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s22, a22, t22, u22], ordered directed edges=[[s22, a22], [a22, s22], [a22, t22], [t22, a22]], distinct source=s22, sink=t22, aligned capacities=[115, 115, 115, 115] and candidate flows=[92, 46, 69, 23]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], [[a22, 115, 115, Yes], [u22, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], [[a22, 115, 115, Yes], [u22, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0074","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"864cdb6969d02bdc3d1520d4babbf3da906b2c714b3b95cb4521e73e580b424b","family_id":"qf-mod44-flow-conservation-audit","seed":202611137074,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[120,1],[120,1],[120,1],[120,1]],\"edges\":[[\"a23\",\"s23\"],[\"a23\",\"t23\"],[\"s23\",\"a23\"],[\"t23\",\"a23\"]],\"flows\":[[48,1],[48,1],[96,1],[24,1]],\"sink\":\"t23\",\"source\":\"s23\",\"vertices\":[\"a23\",\"s23\",\"t23\",\"u23\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s23, a23, t23, u23], ordered directed edges=[[s23, a23], [a23, s23], [a23, t23], [t23, a23]], distinct source=s23, sink=t23, aligned capacities=[120, 120, 120, 120] and candidate flows=[96, 48, 48, 24]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 48], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], [[a23, 120, 96, No], [u23, 0, 0, Yes]], No, [a23, 120, 96, No], every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 48], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], [[a23, 120, 96, No], [u23, 0, 0, Yes]], No, [a23, 120, 96, No], every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0075","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35fc187f1d37ef53c19132b6f8fd7142676487e838d0cffd2b2a9e9016a84ce3","family_id":"qf-mod44-flow-conservation-audit","seed":202611137075,"semantic_template":"qf-mod44-flow-conservation-audit/{\"capacities\":[[125,1],[125,1],[125,1],[125,1]],\"edges\":[[\"a24\",\"s24\"],[\"a24\",\"t24\"],[\"s24\",\"a24\"],[\"t24\",\"a24\"]],\"flows\":[[50,1],[75,1],[100,1],[25,1]],\"sink\":\"t24\",\"source\":\"s24\",\"vertices\":[\"a24\",\"s24\",\"t24\",\"u24\"]}"},"fields":{"title":"Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check","stem":"Use vertices=[s24, a24, t24, u24], ordered directed edges=[[s24, a24], [a24, s24], [a24, t24], [t24, a24]], distinct source=s24, sink=t24, aligned capacities=[125, 125, 125, 125] and candidate flows=[100, 50, 75, 25]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).","answer":{"kind":"expression","value":"(s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, 100], [[t24, a24], 125, 25]], [[a24, 125, 125, Yes], [u24, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)","explanation":"Report [vertex,inflow,outflow,equal] for every vertex except source/sink, including isolated zero-balance vertices. Opposite directed edges contribute according to their own direction. First failed vertex is separate from global net-value equality and from edge capacities. The exact result, retaining the requested order, is (s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, 100], [[t24, a24], 125, 25]], [[a24, 125, 125, Yes], [u24, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)."},"skills":["Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check"],"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-20261113-r07-finite-flow-network-0076","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dfd238c70e97822b0926f17a6f3741c70df57dda806d67aa456be557e3ceb63","family_id":"qf-mod44-flow-value-cross-check","seed":202611137076,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[5,1],[5,1],[5,1],[5,1]],\"edges\":[[\"a0\",\"s0\"],[\"a0\",\"t0\"],[\"s0\",\"a0\"],[\"t0\",\"a0\"]],\"flows\":[[2,1],[3,1],[4,1],[1,1]],\"sink\":\"t0\",\"source\":\"s0\",\"vertices\":[\"a0\",\"s0\",\"t0\",\"u0\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s0, a0, t0, u0], ordered directed edges=[[s0, a0], [a0, s0], [a0, t0], [t0, a0]], distinct source=s0, sink=t0, aligned capacities=[5, 5, 5, 5] and candidate flows=[4, 2, 3, 1]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, 4], [[t0, a0], 5, 1]], 2, 2, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s0, t0, [[[a0, s0], 5, 2], [[a0, t0], 5, 3], [[s0, a0], 5, 4], [[t0, a0], 5, 1]], 2, 2, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0077","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63cf16b72a24d6d03891ab0a24e457e45c03b3895a895155510008de9ed1e356","family_id":"qf-mod44-flow-value-cross-check","seed":202611137077,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[10,1],[10,1],[10,1],[10,1]],\"edges\":[[\"a1\",\"s1\"],[\"a1\",\"t1\"],[\"s1\",\"a1\"],[\"t1\",\"a1\"]],\"flows\":[[4,1],[6,1],[8,1],[2,1]],\"sink\":\"t1\",\"source\":\"s1\",\"vertices\":[\"a1\",\"s1\",\"t1\",\"u1\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s1, a1, t1, u1], ordered directed edges=[[s1, a1], [a1, s1], [a1, t1], [t1, a1]], distinct source=s1, sink=t1, aligned capacities=[10, 10, 10, 10] and candidate flows=[8, 4, 6, 2]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], 4, 4, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], 4, 4, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0078","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30fca348d88d4fe96b5e3e45dfdcb40d73d47ddfe7d9fd5b4cfb9366c039b60b","family_id":"qf-mod44-flow-value-cross-check","seed":202611137078,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[15,1],[15,1],[15,1],[15,1]],\"edges\":[[\"a2\",\"s2\"],[\"a2\",\"t2\"],[\"s2\",\"a2\"],[\"t2\",\"a2\"]],\"flows\":[[6,1],[9,1],[12,1],[3,1]],\"sink\":\"t2\",\"source\":\"s2\",\"vertices\":[\"a2\",\"s2\",\"t2\",\"u2\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s2, a2, t2, u2], ordered directed edges=[[s2, a2], [a2, s2], [a2, t2], [t2, a2]], distinct source=s2, sink=t2, aligned capacities=[15, 15, 15, 15] and candidate flows=[12, 6, 9, 3]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], 6, 6, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s2, t2, [[[a2, s2], 15, 6], [[a2, t2], 15, 9], [[s2, a2], 15, 12], [[t2, a2], 15, 3]], 6, 6, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0079","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f13a7ff9d06462a7f58254a11021b914176bf55a3b515dbb54bf2aa8bf60c150","family_id":"qf-mod44-flow-value-cross-check","seed":202611137079,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[20,1],[20,1],[20,1],[20,1]],\"edges\":[[\"a3\",\"s3\"],[\"a3\",\"t3\"],[\"s3\",\"a3\"],[\"t3\",\"a3\"]],\"flows\":[[8,1],[12,1],[16,1],[4,1]],\"sink\":\"t3\",\"source\":\"s3\",\"vertices\":[\"a3\",\"s3\",\"t3\",\"u3\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s3, a3, t3, u3], ordered directed edges=[[s3, a3], [a3, s3], [a3, t3], [t3, a3]], distinct source=s3, sink=t3, aligned capacities=[20, 20, 20, 20] and candidate flows=[16, 8, 12, 4]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, 16], [[t3, a3], 20, 4]], 8, 8, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, 16], [[t3, a3], 20, 4]], 8, 8, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0080","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6908ec8691b16c763aa147e4209ba210dd123c68ce10a2e528758efc445f76bd","family_id":"qf-mod44-flow-value-cross-check","seed":202611137080,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[25,1],[25,1],[25,1],[25,1]],\"edges\":[[\"a4\",\"s4\"],[\"a4\",\"t4\"],[\"s4\",\"a4\"],[\"t4\",\"a4\"]],\"flows\":[[10,1],[15,1],[20,1],[5,1]],\"sink\":\"t4\",\"source\":\"s4\",\"vertices\":[\"a4\",\"s4\",\"t4\",\"u4\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s4, a4, t4, u4], ordered directed edges=[[s4, a4], [a4, s4], [a4, t4], [t4, a4]], distinct source=s4, sink=t4, aligned capacities=[25, 25, 25, 25] and candidate flows=[20, 10, 15, 5]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 20], [[t4, a4], 25, 5]], 10, 10, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s4, t4, [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 20], [[t4, a4], 25, 5]], 10, 10, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0081","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"999f8a8b423cfe3c8b4860c9bd121c4d6c1c41e403bc0da1767276c195797e15","family_id":"qf-mod44-flow-value-cross-check","seed":202611137081,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[30,1],[30,1],[30,1],[30,1]],\"edges\":[[\"a5\",\"s5\"],[\"a5\",\"t5\"],[\"s5\",\"a5\"],[\"t5\",\"a5\"]],\"flows\":[[12,1],[18,1],[24,1],[6,1]],\"sink\":\"t5\",\"source\":\"s5\",\"vertices\":[\"a5\",\"s5\",\"t5\",\"u5\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s5, a5, t5, u5], ordered directed edges=[[s5, a5], [a5, s5], [a5, t5], [t5, a5]], distinct source=s5, sink=t5, aligned capacities=[30, 30, 30, 30] and candidate flows=[24, 12, 18, 6]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], 12, 12, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s5, t5, [[[a5, s5], 30, 12], [[a5, t5], 30, 18], [[s5, a5], 30, 24], [[t5, a5], 30, 6]], 12, 12, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0082","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12a20fc59ef1c6f1b8bfe00405e88530a63a17cc7fe45c453f0702c7dc66c4be","family_id":"qf-mod44-flow-value-cross-check","seed":202611137082,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[35,1],[35,1],[35,1],[35,1]],\"edges\":[[\"a6\",\"s6\"],[\"a6\",\"t6\"],[\"s6\",\"a6\"],[\"t6\",\"a6\"]],\"flows\":[[14,1],[21,1],[28,1],[7,1]],\"sink\":\"t6\",\"source\":\"s6\",\"vertices\":[\"a6\",\"s6\",\"t6\",\"u6\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s6, a6, t6, u6], ordered directed edges=[[s6, a6], [a6, s6], [a6, t6], [t6, a6]], distinct source=s6, sink=t6, aligned capacities=[35, 35, 35, 35] and candidate flows=[28, 14, 21, 7]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], 14, 14, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], 14, 14, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0083","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a35800e2dae456a012f5ffab1ded4834fdae39e8eb77f0aa6454df45977e53fe","family_id":"qf-mod44-flow-value-cross-check","seed":202611137083,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[40,1],[40,1],[40,1],[40,1]],\"edges\":[[\"a7\",\"s7\"],[\"a7\",\"t7\"],[\"s7\",\"a7\"],[\"t7\",\"a7\"]],\"flows\":[[16,1],[24,1],[32,1],[8,1]],\"sink\":\"t7\",\"source\":\"s7\",\"vertices\":[\"a7\",\"s7\",\"t7\",\"u7\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s7, a7, t7, u7], ordered directed edges=[[s7, a7], [a7, s7], [a7, t7], [t7, a7]], distinct source=s7, sink=t7, aligned capacities=[40, 40, 40, 40] and candidate flows=[32, 16, 24, 8]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 24], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], 16, 16, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s7, t7, [[[a7, s7], 40, 16], [[a7, t7], 40, 24], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], 16, 16, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0084","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f495c61fc637fe58bba7e5042a5c64704e603ccd56b90b5cd7384c94d53f1a1","family_id":"qf-mod44-flow-value-cross-check","seed":202611137084,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[45,1],[45,1],[45,1],[45,1]],\"edges\":[[\"a8\",\"s8\"],[\"a8\",\"t8\"],[\"s8\",\"a8\"],[\"t8\",\"a8\"]],\"flows\":[[18,1],[27,1],[36,1],[9,1]],\"sink\":\"t8\",\"source\":\"s8\",\"vertices\":[\"a8\",\"s8\",\"t8\",\"u8\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s8, a8, t8, u8], ordered directed edges=[[s8, a8], [a8, s8], [a8, t8], [t8, a8]], distinct source=s8, sink=t8, aligned capacities=[45, 45, 45, 45] and candidate flows=[36, 18, 27, 9]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], 18, 18, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s8, t8, [[[a8, s8], 45, 18], [[a8, t8], 45, 27], [[s8, a8], 45, 36], [[t8, a8], 45, 9]], 18, 18, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0085","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afe20e99570951461cd5d80bc538921e13c8e38652cdac506f5120bc0f49983c","family_id":"qf-mod44-flow-value-cross-check","seed":202611137085,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[50,1],[50,1],[50,1],[50,1]],\"edges\":[[\"a9\",\"s9\"],[\"a9\",\"t9\"],[\"s9\",\"a9\"],[\"t9\",\"a9\"]],\"flows\":[[20,1],[30,1],[40,1],[10,1]],\"sink\":\"t9\",\"source\":\"s9\",\"vertices\":[\"a9\",\"s9\",\"t9\",\"u9\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s9, a9, t9, u9], ordered directed edges=[[s9, a9], [a9, s9], [a9, t9], [t9, a9]], distinct source=s9, sink=t9, aligned capacities=[50, 50, 50, 50] and candidate flows=[40, 20, 30, 10]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, 40], [[t9, a9], 50, 10]], 20, 20, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s9, t9, [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, 40], [[t9, a9], 50, 10]], 20, 20, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0086","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b63c7ccd75dbaa4c22375d7805f48e93653c7d71d77b2a8c2426aa4d4afea81a","family_id":"qf-mod44-flow-value-cross-check","seed":202611137086,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[55,1],[55,1],[55,1],[55,1]],\"edges\":[[\"a10\",\"s10\"],[\"a10\",\"t10\"],[\"s10\",\"a10\"],[\"t10\",\"a10\"]],\"flows\":[[22,1],[33,1],[44,1],[11,1]],\"sink\":\"t10\",\"source\":\"s10\",\"vertices\":[\"a10\",\"s10\",\"t10\",\"u10\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s10, a10, t10, u10], ordered directed edges=[[s10, a10], [a10, s10], [a10, t10], [t10, a10]], distinct source=s10, sink=t10, aligned capacities=[55, 55, 55, 55] and candidate flows=[44, 22, 33, 11]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 44], [[t10, a10], 55, 11]], 22, 22, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s10, t10, [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 44], [[t10, a10], 55, 11]], 22, 22, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0087","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bcc6a62598b48b2c84d4dfc901e0d9f72dc0957b33ebf3522168a8444d9ccc4","family_id":"qf-mod44-flow-value-cross-check","seed":202611137087,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[60,1],[60,1],[60,1],[60,1]],\"edges\":[[\"a11\",\"s11\"],[\"a11\",\"t11\"],[\"s11\",\"a11\"],[\"t11\",\"a11\"]],\"flows\":[[24,1],[36,1],[48,1],[12,1]],\"sink\":\"t11\",\"source\":\"s11\",\"vertices\":[\"a11\",\"s11\",\"t11\",\"u11\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s11, a11, t11, u11], ordered directed edges=[[s11, a11], [a11, s11], [a11, t11], [t11, a11]], distinct source=s11, sink=t11, aligned capacities=[60, 60, 60, 60] and candidate flows=[48, 24, 36, 12]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 36], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], 24, 24, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s11, t11, [[[a11, s11], 60, 24], [[a11, t11], 60, 36], [[s11, a11], 60, 48], [[t11, a11], 60, 12]], 24, 24, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0088","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03a3ea19d9c51e96ecd414e07dd3c589f827399e029603cdf458fb546d0ed498","family_id":"qf-mod44-flow-value-cross-check","seed":202611137088,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[65,1],[65,1],[65,1],[65,1]],\"edges\":[[\"a12\",\"s12\"],[\"a12\",\"t12\"],[\"s12\",\"a12\"],[\"t12\",\"a12\"]],\"flows\":[[26,1],[39,1],[52,1],[13,1]],\"sink\":\"t12\",\"source\":\"s12\",\"vertices\":[\"a12\",\"s12\",\"t12\",\"u12\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s12, a12, t12, u12], ordered directed edges=[[s12, a12], [a12, s12], [a12, t12], [t12, a12]], distinct source=s12, sink=t12, aligned capacities=[65, 65, 65, 65] and candidate flows=[52, 26, 39, 13]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, 52], [[t12, a12], 65, 13]], 26, 26, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s12, t12, [[[a12, s12], 65, 26], [[a12, t12], 65, 39], [[s12, a12], 65, 52], [[t12, a12], 65, 13]], 26, 26, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0089","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be5052343a1c93e66247ec2081a066d843b8508ba7569926416889331fae61bb","family_id":"qf-mod44-flow-value-cross-check","seed":202611137089,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[70,1],[70,1],[70,1],[70,1]],\"edges\":[[\"a13\",\"s13\"],[\"a13\",\"t13\"],[\"s13\",\"a13\"],[\"t13\",\"a13\"]],\"flows\":[[28,1],[42,1],[56,1],[14,1]],\"sink\":\"t13\",\"source\":\"s13\",\"vertices\":[\"a13\",\"s13\",\"t13\",\"u13\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s13, a13, t13, u13], ordered directed edges=[[s13, a13], [a13, s13], [a13, t13], [t13, a13]], distinct source=s13, sink=t13, aligned capacities=[70, 70, 70, 70] and candidate flows=[56, 28, 42, 14]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 42], [[s13, a13], 70, 56], [[t13, a13], 70, 14]], 28, 28, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s13, t13, [[[a13, s13], 70, 28], [[a13, t13], 70, 42], [[s13, a13], 70, 56], [[t13, a13], 70, 14]], 28, 28, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0090","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cd6a783efd3bb1263719a482d00b91150bb6f1204cc852501c6a9410644c5da","family_id":"qf-mod44-flow-value-cross-check","seed":202611137090,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[75,1],[75,1],[75,1],[75,1]],\"edges\":[[\"a14\",\"s14\"],[\"a14\",\"t14\"],[\"s14\",\"a14\"],[\"t14\",\"a14\"]],\"flows\":[[30,1],[45,1],[60,1],[15,1]],\"sink\":\"t14\",\"source\":\"s14\",\"vertices\":[\"a14\",\"s14\",\"t14\",\"u14\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s14, a14, t14, u14], ordered directed edges=[[s14, a14], [a14, s14], [a14, t14], [t14, a14]], distinct source=s14, sink=t14, aligned capacities=[75, 75, 75, 75] and candidate flows=[60, 30, 45, 15]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], 30, 30, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s14, t14, [[[a14, s14], 75, 30], [[a14, t14], 75, 45], [[s14, a14], 75, 60], [[t14, a14], 75, 15]], 30, 30, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0091","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2874124ee304bef1589d800b2a6069909f809d230cc0e94fd93a12822fdb591e","family_id":"qf-mod44-flow-value-cross-check","seed":202611137091,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[80,1],[80,1],[80,1],[80,1]],\"edges\":[[\"a15\",\"s15\"],[\"a15\",\"t15\"],[\"s15\",\"a15\"],[\"t15\",\"a15\"]],\"flows\":[[32,1],[48,1],[64,1],[16,1]],\"sink\":\"t15\",\"source\":\"s15\",\"vertices\":[\"a15\",\"s15\",\"t15\",\"u15\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s15, a15, t15, u15], ordered directed edges=[[s15, a15], [a15, s15], [a15, t15], [t15, a15]], distinct source=s15, sink=t15, aligned capacities=[80, 80, 80, 80] and candidate flows=[64, 32, 48, 16]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, 64], [[t15, a15], 80, 16]], 32, 32, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s15, t15, [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, 64], [[t15, a15], 80, 16]], 32, 32, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0092","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5aecbaa565f89b9e0bc38eb5c7aabcc417265b69d5382b33a95a66038ae0f203","family_id":"qf-mod44-flow-value-cross-check","seed":202611137092,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[85,1],[85,1],[85,1],[85,1]],\"edges\":[[\"a16\",\"s16\"],[\"a16\",\"t16\"],[\"s16\",\"a16\"],[\"t16\",\"a16\"]],\"flows\":[[34,1],[51,1],[68,1],[17,1]],\"sink\":\"t16\",\"source\":\"s16\",\"vertices\":[\"a16\",\"s16\",\"t16\",\"u16\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s16, a16, t16, u16], ordered directed edges=[[s16, a16], [a16, s16], [a16, t16], [t16, a16]], distinct source=s16, sink=t16, aligned capacities=[85, 85, 85, 85] and candidate flows=[68, 34, 51, 17]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 68], [[t16, a16], 85, 17]], 34, 34, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s16, t16, [[[a16, s16], 85, 34], [[a16, t16], 85, 51], [[s16, a16], 85, 68], [[t16, a16], 85, 17]], 34, 34, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0093","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca932d94ee2e1a4f4e79906f9afcc6c80b609904290a4a95c0620f06331e793d","family_id":"qf-mod44-flow-value-cross-check","seed":202611137093,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[90,1],[90,1],[90,1],[90,1]],\"edges\":[[\"a17\",\"s17\"],[\"a17\",\"t17\"],[\"s17\",\"a17\"],[\"t17\",\"a17\"]],\"flows\":[[36,1],[54,1],[72,1],[18,1]],\"sink\":\"t17\",\"source\":\"s17\",\"vertices\":[\"a17\",\"s17\",\"t17\",\"u17\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s17, a17, t17, u17], ordered directed edges=[[s17, a17], [a17, s17], [a17, t17], [t17, a17]], distinct source=s17, sink=t17, aligned capacities=[90, 90, 90, 90] and candidate flows=[72, 36, 54, 18]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 54], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], 36, 36, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s17, t17, [[[a17, s17], 90, 36], [[a17, t17], 90, 54], [[s17, a17], 90, 72], [[t17, a17], 90, 18]], 36, 36, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0094","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9f98831ee2f5f0b31d0a879a5bb5dc994723e6bce1d7a47feca7caeaab9f3fb","family_id":"qf-mod44-flow-value-cross-check","seed":202611137094,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[95,1],[95,1],[95,1],[95,1]],\"edges\":[[\"a18\",\"s18\"],[\"a18\",\"t18\"],[\"s18\",\"a18\"],[\"t18\",\"a18\"]],\"flows\":[[38,1],[57,1],[76,1],[19,1]],\"sink\":\"t18\",\"source\":\"s18\",\"vertices\":[\"a18\",\"s18\",\"t18\",\"u18\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s18, a18, t18, u18], ordered directed edges=[[s18, a18], [a18, s18], [a18, t18], [t18, a18]], distinct source=s18, sink=t18, aligned capacities=[95, 95, 95, 95] and candidate flows=[76, 38, 57, 19]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, 76], [[t18, a18], 95, 19]], 38, 38, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s18, t18, [[[a18, s18], 95, 38], [[a18, t18], 95, 57], [[s18, a18], 95, 76], [[t18, a18], 95, 19]], 38, 38, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0095","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21098a252b1b85630174ef97f19e0468e2f71e7585740194e97cdef27bde3e74","family_id":"qf-mod44-flow-value-cross-check","seed":202611137095,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[100,1],[100,1],[100,1],[100,1]],\"edges\":[[\"a19\",\"s19\"],[\"a19\",\"t19\"],[\"s19\",\"a19\"],[\"t19\",\"a19\"]],\"flows\":[[40,1],[60,1],[80,1],[20,1]],\"sink\":\"t19\",\"source\":\"s19\",\"vertices\":[\"a19\",\"s19\",\"t19\",\"u19\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s19, a19, t19, u19], ordered directed edges=[[s19, a19], [a19, s19], [a19, t19], [t19, a19]], distinct source=s19, sink=t19, aligned capacities=[100, 100, 100, 100] and candidate flows=[80, 40, 60, 20]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 60], [[s19, a19], 100, 80], [[t19, a19], 100, 20]], 40, 40, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s19, t19, [[[a19, s19], 100, 40], [[a19, t19], 100, 60], [[s19, a19], 100, 80], [[t19, a19], 100, 20]], 40, 40, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0096","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07c1ee9235e9159795eb1fd1ad10dd0b724b468072cdcadfb9a0fa0d21226f15","family_id":"qf-mod44-flow-value-cross-check","seed":202611137096,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[105,1],[105,1],[105,1],[105,1]],\"edges\":[[\"a20\",\"s20\"],[\"a20\",\"t20\"],[\"s20\",\"a20\"],[\"t20\",\"a20\"]],\"flows\":[[42,1],[63,1],[84,1],[21,1]],\"sink\":\"t20\",\"source\":\"s20\",\"vertices\":[\"a20\",\"s20\",\"t20\",\"u20\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s20, a20, t20, u20], ordered directed edges=[[s20, a20], [a20, s20], [a20, t20], [t20, a20]], distinct source=s20, sink=t20, aligned capacities=[105, 105, 105, 105] and candidate flows=[84, 42, 63, 21]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], 42, 42, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], 42, 42, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0097","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb1f15bfd4af8be172987d51233b9892f894fdeb9bb59001c2612f5c8b201569","family_id":"qf-mod44-flow-value-cross-check","seed":202611137097,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[110,1],[110,1],[110,1],[110,1]],\"edges\":[[\"a21\",\"s21\"],[\"a21\",\"t21\"],[\"s21\",\"a21\"],[\"t21\",\"a21\"]],\"flows\":[[44,1],[66,1],[88,1],[22,1]],\"sink\":\"t21\",\"source\":\"s21\",\"vertices\":[\"a21\",\"s21\",\"t21\",\"u21\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s21, a21, t21, u21], ordered directed edges=[[s21, a21], [a21, s21], [a21, t21], [t21, a21]], distinct source=s21, sink=t21, aligned capacities=[110, 110, 110, 110] and candidate flows=[88, 44, 66, 22]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 66], [[s21, a21], 110, 88], [[t21, a21], 110, 22]], 44, 44, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s21, t21, [[[a21, s21], 110, 44], [[a21, t21], 110, 66], [[s21, a21], 110, 88], [[t21, a21], 110, 22]], 44, 44, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0098","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87b3245c3cf16267c2ad84c2548c3cf34a78586b38f7a994a4acb397c687a49f","family_id":"qf-mod44-flow-value-cross-check","seed":202611137098,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[115,1],[115,1],[115,1],[115,1]],\"edges\":[[\"a22\",\"s22\"],[\"a22\",\"t22\"],[\"s22\",\"a22\"],[\"t22\",\"a22\"]],\"flows\":[[46,1],[69,1],[92,1],[23,1]],\"sink\":\"t22\",\"source\":\"s22\",\"vertices\":[\"a22\",\"s22\",\"t22\",\"u22\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s22, a22, t22, u22], ordered directed edges=[[s22, a22], [a22, s22], [a22, t22], [t22, a22]], distinct source=s22, sink=t22, aligned capacities=[115, 115, 115, 115] and candidate flows=[92, 46, 69, 23]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], 46, 46, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], 46, 46, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0099","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e2f96bf20436c6b1712539c49691269bdc12b0c2d9d85a38807e0ff97ec99c7","family_id":"qf-mod44-flow-value-cross-check","seed":202611137099,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[120,1],[120,1],[120,1],[120,1]],\"edges\":[[\"a23\",\"s23\"],[\"a23\",\"t23\"],[\"s23\",\"a23\"],[\"t23\",\"a23\"]],\"flows\":[[48,1],[72,1],[96,1],[24,1]],\"sink\":\"t23\",\"source\":\"s23\",\"vertices\":[\"a23\",\"s23\",\"t23\",\"u23\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s23, a23, t23, u23], ordered directed edges=[[s23, a23], [a23, s23], [a23, t23], [t23, a23]], distinct source=s23, sink=t23, aligned capacities=[120, 120, 120, 120] and candidate flows=[96, 48, 72, 24]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], 48, 48, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s23, t23, [[[a23, s23], 120, 48], [[a23, t23], 120, 72], [[s23, a23], 120, 96], [[t23, a23], 120, 24]], 48, 48, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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-20261113-r07-finite-flow-network-0100","batch_id":"batch-20261015-advance-20261113-r07-finite-flow-network-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1ac6f8646bd1680a1d2dc1af8edae79429f07966c44ee4ac273b123bf90e040","family_id":"qf-mod44-flow-value-cross-check","seed":202611137100,"semantic_template":"qf-mod44-flow-value-cross-check/{\"capacities\":[[125,1],[125,1],[125,1],[125,1]],\"edges\":[[\"a24\",\"s24\"],[\"a24\",\"t24\"],[\"s24\",\"a24\"],[\"t24\",\"a24\"]],\"flows\":[[50,1],[75,1],[100,1],[25,1]],\"sink\":\"t24\",\"source\":\"s24\",\"vertices\":[\"a24\",\"s24\",\"t24\",\"u24\"]}"},"fields":{"title":"Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties","stem":"Use vertices=[s24, a24, t24, u24], ordered directed edges=[[s24, a24], [a24, s24], [a24, t24], [t24, a24]], distinct source=s24, sink=t24, aligned capacities=[125, 125, 125, 125] and candidate flows=[100, 50, 75, 25]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).","answer":{"kind":"expression","value":"(s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, 100], [[t24, a24], 125, 25]], 50, 50, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)","explanation":"Compute source net outflow and sink net inflow, not gross outgoing totals when reverse edges exist. Equal values can coexist with offsetting internal imbalances, so retain capacity, every-vertex conservation and joint feasibility independently. The fixed No withholds a maximum/min-cut certificate, not the possibility that a special flow is maximum. No residual-network or augmenting-path optimality is inferred. The exact result, retaining the requested order, is (s24, t24, [[[a24, s24], 125, 50], [[a24, t24], 125, 75], [[s24, a24], 125, 100], [[t24, a24], 125, 25]], 50, 50, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)."},"skills":["Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties"],"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."}}