# Directed Capacity, Internal Conservation, Net Values and Nonoptimality Boundaries

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261015-advance-20261113-r07-finite-flow-network-foundations · 2026-10-15 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0001/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 2. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0002/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 3. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0003/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 4. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0004/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 5. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0005/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 6. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0006/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 7. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0007/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 8. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0008/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 9. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0009/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 10. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0010/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 11. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0011/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 12. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0012/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 13. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0013/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 14. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0014/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 15. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0015/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 16. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0016/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 17. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0017/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 18. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0018/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 19. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0019/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 20. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0020/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 21. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0021/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 22. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0022/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 23. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0023/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 24. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0024/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 25. Network Signature and Candidate-Flow Feasibility Are Separate Layers

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).

View answer and explanation([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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0025/)

Finite flow network · qf-mod44-flow-network-qualification · candidate





## 26. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0026/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 27. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0027/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 28. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0028/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 29. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0029/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 30. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0030/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 31. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0031/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 32. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0032/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 33. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0033/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 34. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0034/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 35. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0035/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 36. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0036/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 37. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0037/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 38. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0038/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 39. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0039/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 40. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0040/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 41. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0041/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 42. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0042/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 43. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0043/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 44. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0044/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 45. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0045/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 46. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0046/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 47. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0047/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 48. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0048/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 49. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0049/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 50. Every Edge Needs Zero Less Than or Equal to Flow Less Than or Equal to Capacity

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).

edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0050/)

Finite flow network · qf-mod44-flow-capacity-audit · candidate





## 51. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0051/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 52. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0052/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 53. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0053/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 54. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0054/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 55. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0055/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 56. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0056/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 57. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0057/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 58. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0058/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 59. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0059/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 60. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0060/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 61. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0061/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 62. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0062/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 63. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0063/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 64. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0064/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 65. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0065/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 66. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0066/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 67. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0067/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 68. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0068/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 69. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0069/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 70. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0070/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 71. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0071/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 72. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0072/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 73. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0073/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 74. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0074/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 75. Every Internal Vertex Needs Its Own Complete Inflow-Outflow Check

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0075/)

Finite flow network · qf-mod44-flow-conservation-audit · candidate





## 76. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0076/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 77. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0077/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 78. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0078/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 79. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0079/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 80. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0080/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 81. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0081/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 82. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0082/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 83. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0083/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 84. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0084/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 85. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0085/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 86. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0086/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 87. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0087/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 88. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0088/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 89. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0089/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 90. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0090/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 91. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0091/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 92. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0092/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 93. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0093/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 94. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0094/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 95. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0095/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 96. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0096/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 97. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0097/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 98. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0098/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 99. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0099/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate





## 100. Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

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).

Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r07-finite-flow-network-0100/)

Finite flow network · qf-mod44-flow-value-cross-check · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261113-r07-finite-flow-network-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261113-r07-finite-flow-network-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261113-r07-finite-flow-network-foundations/validation-report.json)