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Question 24377: Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0047-24377 · q-20261015-advance-20261113-r07-finite-flow-network-0077

Original question and givens

Use vertices=[s1, a1, t1, u1], ordered directed edges=[[s1, a1], [a1, s1], [a1, t1], [t1, a1]], distinct source=s1, sink=t1, aligned capacities=[10, 10, 10, 10] and candidate flows=[8, 4, 6, 2]. 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=[s1, a1, t1, u1], ordered directed edges=[[s1, a1], [a1, s1], [a1, t1], [t1, a1]], distinct source=s1, sink=t1, aligned capacities=[10, 10, 10, 10] and candidate flows=[8, 4, 6, 2]. 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).

Definition and scope

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.

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 = [[s1, a1, t1, u1], [[s1, a1], [a1, s1], [a1, t1], [t1, a1]], [10, 10, 10, 10], [8, 4, 6, 2], s1, t1]

After

derived = [[a1, s1, t1, u1], [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], 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, [a1, s1], 4, 10]

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, [a1, t1], 6, 10]

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, [s1, a1], 8, 10]

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, [t1, a1], 2, 10]

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 a1 := sum(8, 2)

After

Flow ALL incoming at vertex a1 = 10

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 a1 := sum(4, 6)

After

Flow ALL outgoing at vertex a1 = 10

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 = [a1, [2, 3], [0, 1]]

After

derived = [10, 10, 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 s1 := sum(4)

After

Flow ALL incoming at vertex s1 = 4

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 s1 := sum(8)

After

Flow ALL outgoing at vertex s1 = 8

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 = [s1, [0], [2]]

After

derived = [4, 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.

Step identifiers and dependencies

Rule: Flow_EACH_vertex_complete_directional_incidence · Uses: G001, S011

S013 · calculation

Before

Flow ALL incoming at vertex t1 := sum(6)

After

Flow ALL incoming at vertex t1 = 6

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 t1 := sum(2)

After

Flow ALL outgoing at vertex t1 = 2

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 = [t1, [1], [3]]

After

derived = [6, 2, 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 u1 := sum()

After

Flow ALL incoming at vertex u1 = 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 u1 := sum()

After

Flow ALL outgoing at vertex u1 = 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 = [u1, [], []]

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(8, 4)

After

Flow source NET outflow not gross outgoing = 4

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(6, 2)

After

Flow sink NET inflow not gross incoming = 4

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 = [s1, t1, [[[a1, s1], 4, 10, Yes], [[a1, t1], 6, 10, Yes], [[s1, a1], 8, 10, Yes], [[t1, a1], 2, 10, Yes]], [[a1, 10, 10, Yes], [u1, 0, 0, Yes]], 4, 4]

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

Reverse checks

  • 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 u1 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Flow ALL incoming at vertex u1 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 t1 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Flow ALL incoming at vertex t1 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 s1 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Flow ALL incoming at vertex s1 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 a1 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Flow ALL incoming at vertex a1 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.

Result

(s1, t1, [[[a1, s1], 10, 4], [[a1, t1], 10, 6], [[s1, a1], 10, 8], [[t1, a1], 10, 2]], 4, 4, 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.

This question review and source binding

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 63cf16b72a24d6d03891ab0a24e457e45c03b3895a895155510008de9ed1e356
  • candidate record SHA-256: b677448be0f2c5a17af43e3215f22f6f4f96527b9c02da27d1c9f952ecf70ede
  • semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386
  • question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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