Original question and givens
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).
G001 — 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).
Step-by-step derivation
S001 · implies
Before
source roles = [flow-value-cross-check]
After
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]]
Conditions: none
List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode.
Step identifiers and dependencies
Rule: source_ordered_fields_without_JS · Uses: G001
S002 · implies
Before
source roles = [[s15, a15, t15, u15], [[s15, a15], [a15, s15], [a15, t15], [t15, a15]], [80, 80, 80, 80], [64, 32, 48, 16], s15, t15]
After
derived = [[a15, s15, t15, u15], [[[a15, s15], 80, 32], [[a15, t15], 80, 48], [[s15, a15], 80, 64], [[t15, a15], 80, 16]], 2, Yes]
Conditions: none
Zip edge,capacity and flow by original index before source English canonical-edge sorting;vertices use UTF16.Opposite directions are distinct,zero capacity and isolated vertices are legal,and negative flow is retained for capacity failure rather than rewritten.
Step identifiers and dependencies
Rule: Flow_zip_source_edges_capacities_flows_BEFORE_order · Uses: G001, S001
S003 · implies
Before
source roles = [0, [a15, s15], 32, 80]
After
derived = [Yes, Yes, Yes]
Conditions: none
Every directed edge needs0<=flow<=capacity.A qualified network signature does not imply candidate-flow feasibility.
Step identifiers and dependencies
Rule: Flow_EACH_edge_two_sided_capacity_obligation · Uses: G001, S002
S004 · implies
Before
source roles = [1, [a15, t15], 48, 80]
After
derived = [Yes, Yes, Yes]
Conditions: none
Every directed edge needs0<=flow<=capacity.A qualified network signature does not imply candidate-flow feasibility.
Step identifiers and dependencies
Rule: Flow_EACH_edge_two_sided_capacity_obligation · Uses: G001, S003
S005 · implies
Before
source roles = [2, [s15, a15], 64, 80]
After
derived = [Yes, Yes, Yes]
Conditions: none
Every directed edge needs0<=flow<=capacity.A qualified network signature does not imply candidate-flow feasibility.
Step identifiers and dependencies
Rule: Flow_EACH_edge_two_sided_capacity_obligation · Uses: G001, S004
S006 · implies
Before
source roles = [3, [t15, a15], 16, 80]
After
derived = [Yes, Yes, Yes]
Conditions: none
Every directed edge needs0<=flow<=capacity.A qualified network signature does not imply candidate-flow feasibility.
Step identifiers and dependencies
Rule: Flow_EACH_edge_two_sided_capacity_obligation · Uses: G001, S005
S007 · calculation
Before
Flow ALL incoming at vertex a15 := sum(64, 16)
After
Flow ALL incoming at vertex a15 = 80
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
Flow ALL outgoing at vertex a15 := sum(32, 48)
After
Flow ALL outgoing at vertex a15 = 80
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · implies
Before
source roles = [a15, [2, 3], [0, 1]]
After
derived = [80, 80, Yes, Yes]
Conditions: none
Sum all incoming and outgoing edges separately;an isolated vertex has0=0.Conservation is required only internally,but compute source/sink balances too for net values.
Step identifiers and dependencies
Rule: Flow_EACH_vertex_complete_directional_incidence · Uses: G001, S008
S010 · calculation
Before
Flow ALL incoming at vertex s15 := sum(32)
After
Flow ALL incoming at vertex s15 = 32
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
Flow ALL outgoing at vertex s15 := sum(64)
After
Flow ALL outgoing at vertex s15 = 64
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
source roles = [s15, [0], [2]]
After
derived = [32, 64, No, No]
Conditions: none
Sum all incoming and outgoing edges separately;an isolated vertex has0=0.Conservation is required only internally,but compute source/sink balances too for net values.
Step identifiers and dependencies
Rule: Flow_EACH_vertex_complete_directional_incidence · Uses: G001, S011
S013 · calculation
Before
Flow ALL incoming at vertex t15 := sum(48)
After
Flow ALL incoming at vertex t15 = 48
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · calculation
Before
Flow ALL outgoing at vertex t15 := sum(16)
After
Flow ALL outgoing at vertex t15 = 16
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S013
S015 · implies
Before
source roles = [t15, [1], [3]]
After
derived = [48, 16, No, No]
Conditions: none
Sum all incoming and outgoing edges separately;an isolated vertex has0=0.Conservation is required only internally,but compute source/sink balances too for net values.
Step identifiers and dependencies
Rule: Flow_EACH_vertex_complete_directional_incidence · Uses: G001, S014
S016 · calculation
Before
Flow ALL incoming at vertex u15 := sum()
After
Flow ALL incoming at vertex u15 = 0
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S015
S017 · calculation
Before
Flow ALL outgoing at vertex u15 := sum()
After
Flow ALL outgoing at vertex u15 = 0
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · implies
Before
source roles = [u15, [], []]
After
derived = [0, 0, Yes, Yes]
Conditions: none
Sum all incoming and outgoing edges separately;an isolated vertex has0=0.Conservation is required only internally,but compute source/sink balances too for net values.
Step identifiers and dependencies
Rule: Flow_EACH_vertex_complete_directional_incidence · Uses: G001, S017
S019 · calculation
Before
Flow source NET outflow not gross outgoing := sub(64, 32)
After
Flow source NET outflow not gross outgoing = 32
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · calculation
Before
Flow sink NET inflow not gross incoming := sub(48, 16)
After
Flow sink NET inflow not gross incoming = 32
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S019
S021 · implies
Before
source roles = [s15, t15, [[[a15, s15], 32, 80, Yes], [[a15, t15], 48, 80, Yes], [[s15, a15], 64, 80, Yes], [[t15, a15], 16, 80, Yes]], [[a15, 80, 80, Yes], [u15, 0, 0, Yes]], 32, 32]
After
derived = [Yes, Yes, Yes, Yes, No]
Conditions: none
Every edge cancels in the sum of all vertex balances,so internal conservation implies equal source/sink net values.The converse fails when internal imbalances cancel.Capacity,each balance and net equality remain separate.Feasibility does not certify maximum/min-cut;No does not deny special optimal examples.
Step identifiers and dependencies
Rule: Flow_value_capacity_conservation_and_maximum_separate · Uses: G001, S020
Result
(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)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.