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R0047-24361 · q-20261015-advance-20261113-r07-finite-flow-network-0011
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).
G001 — 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).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 = [[s10, a10, t10, u10], [[s10, a10], [a10, s10], [a10, t10], [t10, a10]], [55, 55, 55, 55], [44, 22, 33, 11], s10, t10]
After
derived = [[a10, s10, t10, u10], [[[a10, s10], 55, 22], [[a10, t10], 55, 33], [[s10, a10], 55, 44], [[t10, a10], 55, 11]], 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, [a10, s10], 22, 55]
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, [a10, t10], 33, 55]
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, [s10, a10], 44, 55]
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, [t10, a10], 11, 55]
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 a10 := sum(44, 11)
After
Flow ALL incoming at vertex a10 = 55
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 a10 := sum(22, 33)
After
Flow ALL outgoing at vertex a10 = 55
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 = [a10, [2, 3], [0, 1]]
After
derived = [55, 55, 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 s10 := sum(22)
After
Flow ALL incoming at vertex s10 = 22
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 s10 := sum(44)
After
Flow ALL outgoing at vertex s10 = 44
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 = [s10, [0], [2]]
After
derived = [22, 44, 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 t10 := sum(33)
After
Flow ALL incoming at vertex t10 = 33
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 t10 := sum(11)
After
Flow ALL outgoing at vertex t10 = 11
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 = [t10, [1], [3]]
After
derived = [33, 11, 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 u10 := sum()
After
Flow ALL incoming at vertex u10 = 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 u10 := sum()
After
Flow ALL outgoing at vertex u10 = 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 = [u10, [], []]
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(44, 22)
After
Flow source NET outflow not gross outgoing = 22
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(33, 11)
After
Flow sink NET inflow not gross incoming = 22
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 = [s10, t10, [[[a10, s10], 22, 55, Yes], [[a10, t10], 33, 55, Yes], [[s10, a10], 44, 55, Yes], [[t10, a10], 11, 55, Yes]], [[a10, 55, 55, Yes], [u10, 0, 0, Yes]], 22, 22]
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 u10 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute Flow ALL incoming at vertex u10 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 t10 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute Flow ALL incoming at vertex t10 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 s10 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute Flow ALL incoming at vertex s10 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 a10 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Flow ALL incoming at vertex a10 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.([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)
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: 24b72f29c5123acc58bc3ea554f06b3c28c120dc91517339d3d784de77fae656candidate record SHA-256: dba333a38dd4265f21e0c506c635a98c8337150fff6b51b056bd101248673ba2semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50bLean: NOT_RUN · human review: NOT_PERFORMED