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R0047-24358 · q-20261015-advance-20261113-r07-finite-flow-network-0008
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).
G001 — 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).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.
Before
source roles = [flow-network-qualification]
After
derived = [[vertices, source, sink, edge_capacity_flow_rows, antiparallel_pair_count, network_qualified, convention]]
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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [[s7, a7, t7, u7], [[s7, a7], [a7, s7], [a7, t7], [t7, a7]], [40, 40, 40, 40], [32, 16, 24, 8], s7, t7]
After
derived = [[a7, s7, t7, u7], [[[a7, s7], 40, 16], [[a7, t7], 40, 24], [[s7, a7], 40, 32], [[t7, a7], 40, 8]], 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.
Rule: Flow_zip_source_edges_capacities_flows_BEFORE_order · Uses: G001, S001
Before
source roles = [0, [a7, s7], 16, 40]
After
derived = [Yes, Yes, Yes]
Conditions: none
Every directed edge needs0<=flow<=capacity.A qualified network signature does not imply candidate-flow feasibility.
Rule: Flow_EACH_edge_two_sided_capacity_obligation · Uses: G001, S002
Before
source roles = [1, [a7, t7], 24, 40]
After
derived = [Yes, Yes, Yes]
Conditions: none
Every directed edge needs0<=flow<=capacity.A qualified network signature does not imply candidate-flow feasibility.
Rule: Flow_EACH_edge_two_sided_capacity_obligation · Uses: G001, S003
Before
source roles = [2, [s7, a7], 32, 40]
After
derived = [Yes, Yes, Yes]
Conditions: none
Every directed edge needs0<=flow<=capacity.A qualified network signature does not imply candidate-flow feasibility.
Rule: Flow_EACH_edge_two_sided_capacity_obligation · Uses: G001, S004
Before
source roles = [3, [t7, a7], 8, 40]
After
derived = [Yes, Yes, Yes]
Conditions: none
Every directed edge needs0<=flow<=capacity.A qualified network signature does not imply candidate-flow feasibility.
Rule: Flow_EACH_edge_two_sided_capacity_obligation · Uses: G001, S005
Before
Flow ALL incoming at vertex a7 := sum(32, 8)
After
Flow ALL incoming at vertex a7 = 40
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Flow ALL outgoing at vertex a7 := sum(16, 24)
After
Flow ALL outgoing at vertex a7 = 40
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
source roles = [a7, [2, 3], [0, 1]]
After
derived = [40, 40, 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.
Rule: Flow_EACH_vertex_complete_directional_incidence · Uses: G001, S008
Before
Flow ALL incoming at vertex s7 := sum(16)
After
Flow ALL incoming at vertex s7 = 16
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
Flow ALL outgoing at vertex s7 := sum(32)
After
Flow ALL outgoing at vertex s7 = 32
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
source roles = [s7, [0], [2]]
After
derived = [16, 32, 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.
Rule: Flow_EACH_vertex_complete_directional_incidence · Uses: G001, S011
Before
Flow ALL incoming at vertex t7 := sum(24)
After
Flow ALL incoming at vertex t7 = 24
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
Flow ALL outgoing at vertex t7 := sum(8)
After
Flow ALL outgoing at vertex t7 = 8
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
source roles = [t7, [1], [3]]
After
derived = [24, 8, 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.
Rule: Flow_EACH_vertex_complete_directional_incidence · Uses: G001, S014
Before
Flow ALL incoming at vertex u7 := sum()
After
Flow ALL incoming at vertex u7 = 0
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
Flow ALL outgoing at vertex u7 := sum()
After
Flow ALL outgoing at vertex u7 = 0
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
source roles = [u7, [], []]
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.
Rule: Flow_EACH_vertex_complete_directional_incidence · Uses: G001, S017
Before
Flow source NET outflow not gross outgoing := sub(32, 16)
After
Flow source NET outflow not gross outgoing = 16
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
Flow sink NET inflow not gross incoming := sub(24, 8)
After
Flow sink NET inflow not gross incoming = 16
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
source roles = [s7, t7, [[[a7, s7], 16, 40, Yes], [[a7, t7], 24, 40, Yes], [[s7, a7], 32, 40, Yes], [[t7, a7], 8, 40, Yes]], [[a7, 40, 40, Yes], [u7, 0, 0, Yes]], 16, 16]
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.
Rule: Flow_value_capacity_conservation_and_maximum_separate · Uses: G001, S020
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 u7 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute Flow ALL incoming at vertex u7 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 t7 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute Flow ALL incoming at vertex t7 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 s7 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute Flow ALL incoming at vertex s7 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 a7 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Flow ALL incoming at vertex a7 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.([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)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: af67c0c7d585c4111416b8d8144bf4904c5a6e72ee8418d5c65fe2d3e781baa8candidate record SHA-256: 2ab1a0e0d270bdf0c73218af5d8632e3cadd1f61a83aa769de34f0ce74d23a3csemantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50bLean: NOT_RUN · human review: NOT_PERFORMED