Question 24380: Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

Original question and givens

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).

• G001 — 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).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-finite-flow-network

Step-by-step derivation

source roles = [flow-value-cross-check]

derived = [[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]]

source roles = [[s4, a4, t4, u4], [[s4, a4], [a4, s4], [a4, t4], [t4, a4]], [25, 25, 25, 25], [20, 10, 15, 5], s4, t4]

derived = [[a4, s4, t4, u4], [[[a4, s4], 25, 10], [[a4, t4], 25, 15], [[s4, a4], 25, 20], [[t4, a4], 25, 5]], 2, Yes]

source roles = [0, [a4, s4], 10, 25]

derived = [Yes, Yes, Yes]

source roles = [1, [a4, t4], 15, 25]

derived = [Yes, Yes, Yes]

source roles = [2, [s4, a4], 20, 25]

derived = [Yes, Yes, Yes]

source roles = [3, [t4, a4], 5, 25]

derived = [Yes, Yes, Yes]

Flow ALL incoming at vertex a4 := sum(20, 5)

Flow ALL incoming at vertex a4 = 25

Flow ALL outgoing at vertex a4 := sum(10, 15)

Flow ALL outgoing at vertex a4 = 25

source roles = [a4, [2, 3], [0, 1]]

derived = [25, 25, Yes, Yes]

Flow ALL incoming at vertex s4 := sum(10)

Flow ALL incoming at vertex s4 = 10

Flow ALL outgoing at vertex s4 := sum(20)

Flow ALL outgoing at vertex s4 = 20

source roles = [s4, [0], [2]]

derived = [10, 20, No, No]

Flow ALL incoming at vertex t4 := sum(15)

Flow ALL incoming at vertex t4 = 15

Flow ALL outgoing at vertex t4 := sum(5)

Flow ALL outgoing at vertex t4 = 5

source roles = [t4, [1], [3]]

derived = [15, 5, No, No]

Flow ALL incoming at vertex u4 := sum()

Flow ALL incoming at vertex u4 = 0

Flow ALL outgoing at vertex u4 := sum()

Flow ALL outgoing at vertex u4 = 0

source roles = [u4, [], []]

derived = [0, 0, Yes, Yes]

Flow source NET outflow not gross outgoing := sub(20, 10)

Flow source NET outflow not gross outgoing = 10

Flow sink NET inflow not gross incoming := sub(15, 5)

Flow sink NET inflow not gross incoming = 10

source roles = [s4, t4, [[[a4, s4], 10, 25, Yes], [[a4, t4], 15, 25, Yes], [[s4, a4], 20, 25, Yes], [[t4, a4], 5, 25, Yes]], [[a4, 25, 25, Yes], [u4, 0, 0, Yes]], 10, 10]

derived = [Yes, Yes, Yes, Yes, No]

Reverse checks

• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute Flow sink NET inflow not gross incoming with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Flow source NET outflow not gross outgoing with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute Flow ALL outgoing at vertex u4 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Flow ALL incoming at vertex u4 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute Flow ALL outgoing at vertex t4 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Flow ALL incoming at vertex t4 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute Flow ALL outgoing at vertex s4 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Flow ALL incoming at vertex s4 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute Flow ALL outgoing at vertex a4 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Flow ALL incoming at vertex a4 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(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)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.