Question 24385: Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

Original question and givens

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).

• G001 — 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).

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 = [[s9, a9, t9, u9], [[s9, a9], [a9, s9], [a9, t9], [t9, a9]], [50, 50, 50, 50], [40, 20, 30, 10], s9, t9]

derived = [[a9, s9, t9, u9], [[[a9, s9], 50, 20], [[a9, t9], 50, 30], [[s9, a9], 50, 40], [[t9, a9], 50, 10]], 2, Yes]

source roles = [0, [a9, s9], 20, 50]

derived = [Yes, Yes, Yes]

source roles = [1, [a9, t9], 30, 50]

derived = [Yes, Yes, Yes]

source roles = [2, [s9, a9], 40, 50]

derived = [Yes, Yes, Yes]

source roles = [3, [t9, a9], 10, 50]

derived = [Yes, Yes, Yes]

Flow ALL incoming at vertex a9 := sum(40, 10)

Flow ALL incoming at vertex a9 = 50

Flow ALL outgoing at vertex a9 := sum(20, 30)

Flow ALL outgoing at vertex a9 = 50

source roles = [a9, [2, 3], [0, 1]]

derived = [50, 50, Yes, Yes]

Flow ALL incoming at vertex s9 := sum(20)

Flow ALL incoming at vertex s9 = 20

Flow ALL outgoing at vertex s9 := sum(40)

Flow ALL outgoing at vertex s9 = 40

source roles = [s9, [0], [2]]

derived = [20, 40, No, No]

Flow ALL incoming at vertex t9 := sum(30)

Flow ALL incoming at vertex t9 = 30

Flow ALL outgoing at vertex t9 := sum(10)

Flow ALL outgoing at vertex t9 = 10

source roles = [t9, [1], [3]]

derived = [30, 10, No, No]

Flow ALL incoming at vertex u9 := sum()

Flow ALL incoming at vertex u9 = 0

Flow ALL outgoing at vertex u9 := sum()

Flow ALL outgoing at vertex u9 = 0

source roles = [u9, [], []]

derived = [0, 0, Yes, Yes]

Flow source NET outflow not gross outgoing := sub(40, 20)

Flow source NET outflow not gross outgoing = 20

Flow sink NET inflow not gross incoming := sub(30, 10)

Flow sink NET inflow not gross incoming = 20

source roles = [s9, t9, [[[a9, s9], 20, 50, Yes], [[a9, t9], 30, 50, Yes], [[s9, a9], 40, 50, Yes], [[t9, a9], 10, 50, Yes]], [[a9, 50, 50, Yes], [u9, 0, 0, Yes]], 20, 20]

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 u9 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Flow ALL incoming at vertex u9 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 t9 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Flow ALL incoming at vertex t9 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 s9 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Flow ALL incoming at vertex s9 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 a9 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Flow ALL incoming at vertex a9 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

(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)

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.