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R0047-24346 · q-20261015-advance-20261113-r07-finite-flow-network-0071
Use vertices=[s20, a20, t20, u20], ordered directed edges=[[s20, a20], [a20, s20], [a20, t20], [t20, a20]], distinct source=s20, sink=t20, aligned capacities=[105, 105, 105, 105] and candidate flows=[84, 42, 63, 21]. 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=[s20, a20, t20, u20], ordered directed edges=[[s20, a20], [a20, s20], [a20, t20], [t20, a20]], distinct source=s20, sink=t20, aligned capacities=[105, 105, 105, 105] and candidate flows=[84, 42, 63, 21]. 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 = [[s20, a20, t20, u20], [[s20, a20], [a20, s20], [a20, t20], [t20, a20]], [105, 105, 105, 105], [84, 42, 63, 21], s20, t20]
After
derived = [[a20, s20, t20, u20], [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], 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, [a20, s20], 42, 105]
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, [a20, t20], 63, 105]
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, [s20, a20], 84, 105]
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, [t20, a20], 21, 105]
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 a20 := sum(84, 21)
After
Flow ALL incoming at vertex a20 = 105
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 a20 := sum(42, 63)
After
Flow ALL outgoing at vertex a20 = 105
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 = [a20, [2, 3], [0, 1]]
After
derived = [105, 105, 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 s20 := sum(42)
After
Flow ALL incoming at vertex s20 = 42
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 s20 := sum(84)
After
Flow ALL outgoing at vertex s20 = 84
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 = [s20, [0], [2]]
After
derived = [42, 84, 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 t20 := sum(63)
After
Flow ALL incoming at vertex t20 = 63
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 t20 := sum(21)
After
Flow ALL outgoing at vertex t20 = 21
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 = [t20, [1], [3]]
After
derived = [63, 21, 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 u20 := sum()
After
Flow ALL incoming at vertex u20 = 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 u20 := sum()
After
Flow ALL outgoing at vertex u20 = 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 = [u20, [], []]
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(84, 42)
After
Flow source NET outflow not gross outgoing = 42
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(63, 21)
After
Flow sink NET inflow not gross incoming = 42
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 = [s20, t20, [[[a20, s20], 42, 105, Yes], [[a20, t20], 63, 105, Yes], [[s20, a20], 84, 105, Yes], [[t20, a20], 21, 105, Yes]], [[a20, 105, 105, Yes], [u20, 0, 0, Yes]], 42, 42]
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 u20 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute Flow ALL incoming at vertex u20 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 t20 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute Flow ALL incoming at vertex t20 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 s20 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute Flow ALL incoming at vertex s20 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 a20 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Flow ALL incoming at vertex a20 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.(s20, t20, [[[a20, s20], 105, 42], [[a20, t20], 105, 63], [[s20, a20], 105, 84], [[t20, a20], 105, 21]], [[a20, 105, 105, Yes], [u20, 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: 55958c54c6ddc2dace5753676d56bedaf9667be64daff8e6a24e99a7846093c9candidate record SHA-256: a6d93cae772110976db478dbbb0c71a802f90c8dddefd0947de8ac5688885823semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50bLean: NOT_RUN · human review: NOT_PERFORMED