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R0047-24348 · q-20261015-advance-20261113-r07-finite-flow-network-0073
Use vertices=[s22, a22, t22, u22], ordered directed edges=[[s22, a22], [a22, s22], [a22, t22], [t22, a22]], distinct source=s22, sink=t22, aligned capacities=[115, 115, 115, 115] and candidate flows=[92, 46, 69, 23]. 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, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).
G001 — Use vertices=[s22, a22, t22, u22], ordered directed edges=[[s22, a22], [a22, s22], [a22, t22], [t22, a22]], distinct source=s22, sink=t22, aligned capacities=[115, 115, 115, 115] and candidate flows=[92, 46, 69, 23]. 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, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate).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-conservation-audit]
After
derived = [[source, sink, edge_capacity_flow_rows, internal_balance_rows, conservation_feasible, first_failed_vertex_or_null, certificate]]
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 = [[s22, a22, t22, u22], [[s22, a22], [a22, s22], [a22, t22], [t22, a22]], [115, 115, 115, 115], [92, 46, 69, 23], s22, t22]
After
derived = [[a22, s22, t22, u22], [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], 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, [a22, s22], 46, 115]
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, [a22, t22], 69, 115]
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, [s22, a22], 92, 115]
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, [t22, a22], 23, 115]
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 a22 := sum(92, 23)
After
Flow ALL incoming at vertex a22 = 115
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 a22 := sum(46, 69)
After
Flow ALL outgoing at vertex a22 = 115
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 = [a22, [2, 3], [0, 1]]
After
derived = [115, 115, 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 s22 := sum(46)
After
Flow ALL incoming at vertex s22 = 46
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 s22 := sum(92)
After
Flow ALL outgoing at vertex s22 = 92
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 = [s22, [0], [2]]
After
derived = [46, 92, 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 t22 := sum(69)
After
Flow ALL incoming at vertex t22 = 69
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 t22 := sum(23)
After
Flow ALL outgoing at vertex t22 = 23
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 = [t22, [1], [3]]
After
derived = [69, 23, 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 u22 := sum()
After
Flow ALL incoming at vertex u22 = 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 u22 := sum()
After
Flow ALL outgoing at vertex u22 = 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 = [u22, [], []]
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(92, 46)
After
Flow source NET outflow not gross outgoing = 46
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(69, 23)
After
Flow sink NET inflow not gross incoming = 46
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 = [s22, t22, [[[a22, s22], 46, 115, Yes], [[a22, t22], 69, 115, Yes], [[s22, a22], 92, 115, Yes], [[t22, a22], 23, 115, Yes]], [[a22, 115, 115, Yes], [u22, 0, 0, Yes]], 46, 46]
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 u22 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute Flow ALL incoming at vertex u22 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 t22 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute Flow ALL incoming at vertex t22 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 s22 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute Flow ALL incoming at vertex s22 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 a22 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Flow ALL incoming at vertex a22 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.(s22, t22, [[[a22, s22], 115, 46], [[a22, t22], 115, 69], [[s22, a22], 115, 92], [[t22, a22], 115, 23]], [[a22, 115, 115, Yes], [u22, 0, 0, Yes]], Yes, null, every_internal_vertex_requires_complete_inflow_equals_outflow)
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: dd1ee89153dcf01002b5a553925e1eb4f149ee28c62c23142bd8cd1938ea82cbcandidate record SHA-256: 8c0f8d0bc28a3d0ccd9bb9123244e1cfcd9d5c2ea56e46f9ca0b03c9d2781365semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50bLean: NOT_RUN · human review: NOT_PERFORMED