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Question 25940: An Equal-Valued Feasible Flow and Cut Certify Both Optima

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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R0050-25940 · q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0090

Original question and givens

Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[16, 16, 17, 17, 18, 18, 19, 19], candidate flows=[16, 16, 17, 17, 18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).

  • G001 — Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[16, 16, 17, 17, 18, 18, 19, 19], candidate flows=[16, 16, 17, 17, 18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).

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 = [mfmc-equal-value-optimality]

After

derived = [[flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate]]

Conditions: none

List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [[s, a0, a1, a2, a3, t], [[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], [16, 16, 17, 17, 18, 18, 19, 19], [16, 16, 17, 17, 18, 18, 19, 19], s, t]

After

derived = [[a0, a1, a2, a3, s, t], [[a0, t], [a1, t], [a2, t], [a3, t], [s, a0], [s, a1], [s, a2], [s, a3]], [16, 17, 18, 19, 16, 17, 18, 19], [16, 17, 18, 19, 16, 17, 18, 19]]

Conditions: none

Use2..12 vertices.Zip directed edges with capacity/flow by original index before sorting;reverse arcs remain distinct.Source and sink differ;reject loops/duplicates but allow zero capacities,no edges and isolated vertices.

Step identifiers and dependencies

Rule: MFMC_complete_directed_network_ZIP_before_source_order · Uses: G001, S001

S003 · calculation

Before

MFMC cut mask0 OUTWARD capacity := sum(16, 17, 18, 19)

After

MFMC cut mask0 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [0, [a0, a1, a2, a3], [s]]

After

derived = [[4, 5, 6, 7], [], [[[s, a0], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S003

S005 · calculation

Before

MFMC cut mask1 OUTWARD capacity := sum(16, 17, 18, 19)

After

MFMC cut mask1 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [1, [a0, a1, a2, a3], [a0, s]]

After

derived = [[0, 5, 6, 7], [], [[[a0, t], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S005

S007 · calculation

Before

MFMC cut mask2 OUTWARD capacity := sum(17, 16, 18, 19)

After

MFMC cut mask2 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

source roles = [2, [a0, a1, a2, a3], [a1, s]]

After

derived = [[1, 4, 6, 7], [], [[[a1, t], 17], [[s, a0], 16], [[s, a2], 18], [[s, a3], 19]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S007

S009 · calculation

Before

MFMC cut mask3 OUTWARD capacity := sum(16, 17, 18, 19)

After

MFMC cut mask3 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [3, [a0, a1, a2, a3], [a0, a1, s]]

After

derived = [[0, 1, 6, 7], [], [[[a0, t], 16], [[a1, t], 17], [[s, a2], 18], [[s, a3], 19]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S009

S011 · calculation

Before

MFMC cut mask4 OUTWARD capacity := sum(18, 16, 17, 19)

After

MFMC cut mask4 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

source roles = [4, [a0, a1, a2, a3], [a2, s]]

After

derived = [[2, 4, 5, 7], [], [[[a2, t], 18], [[s, a0], 16], [[s, a1], 17], [[s, a3], 19]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S011

S013 · calculation

Before

MFMC cut mask5 OUTWARD capacity := sum(16, 18, 17, 19)

After

MFMC cut mask5 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

source roles = [5, [a0, a1, a2, a3], [a0, a2, s]]

After

derived = [[0, 2, 5, 7], [], [[[a0, t], 16], [[a2, t], 18], [[s, a1], 17], [[s, a3], 19]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S013

S015 · calculation

Before

MFMC cut mask6 OUTWARD capacity := sum(17, 18, 16, 19)

After

MFMC cut mask6 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

source roles = [6, [a0, a1, a2, a3], [a1, a2, s]]

After

derived = [[1, 2, 4, 7], [], [[[a1, t], 17], [[a2, t], 18], [[s, a0], 16], [[s, a3], 19]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S015

S017 · calculation

Before

MFMC cut mask7 OUTWARD capacity := sum(16, 17, 18, 19)

After

MFMC cut mask7 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

source roles = [7, [a0, a1, a2, a3], [a0, a1, a2, s]]

After

derived = [[0, 1, 2, 7], [], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[s, a3], 19]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S017

S019 · calculation

Before

MFMC cut mask8 OUTWARD capacity := sum(19, 16, 17, 18)

After

MFMC cut mask8 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

source roles = [8, [a0, a1, a2, a3], [a3, s]]

After

derived = [[3, 4, 5, 6], [], [[[a3, t], 19], [[s, a0], 16], [[s, a1], 17], [[s, a2], 18]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S019

S021 · calculation

Before

MFMC cut mask9 OUTWARD capacity := sum(16, 19, 17, 18)

After

MFMC cut mask9 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

source roles = [9, [a0, a1, a2, a3], [a0, a3, s]]

After

derived = [[0, 3, 5, 6], [], [[[a0, t], 16], [[a3, t], 19], [[s, a1], 17], [[s, a2], 18]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S021

S023 · calculation

Before

MFMC cut mask10 OUTWARD capacity := sum(17, 19, 16, 18)

After

MFMC cut mask10 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

source roles = [10, [a0, a1, a2, a3], [a1, a3, s]]

After

derived = [[1, 3, 4, 6], [], [[[a1, t], 17], [[a3, t], 19], [[s, a0], 16], [[s, a2], 18]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S023

S025 · calculation

Before

MFMC cut mask11 OUTWARD capacity := sum(16, 17, 19, 18)

After

MFMC cut mask11 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

source roles = [11, [a0, a1, a2, a3], [a0, a1, a3, s]]

After

derived = [[0, 1, 3, 6], [], [[[a0, t], 16], [[a1, t], 17], [[a3, t], 19], [[s, a2], 18]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S025

S027 · calculation

Before

MFMC cut mask12 OUTWARD capacity := sum(18, 19, 16, 17)

After

MFMC cut mask12 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

source roles = [12, [a0, a1, a2, a3], [a2, a3, s]]

After

derived = [[2, 3, 4, 5], [], [[[a2, t], 18], [[a3, t], 19], [[s, a0], 16], [[s, a1], 17]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S027

S029 · calculation

Before

MFMC cut mask13 OUTWARD capacity := sum(16, 18, 19, 17)

After

MFMC cut mask13 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

source roles = [13, [a0, a1, a2, a3], [a0, a2, a3, s]]

After

derived = [[0, 2, 3, 5], [], [[[a0, t], 16], [[a2, t], 18], [[a3, t], 19], [[s, a1], 17]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S029

S031 · calculation

Before

MFMC cut mask14 OUTWARD capacity := sum(17, 18, 19, 16)

After

MFMC cut mask14 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · implies

Before

source roles = [14, [a0, a1, a2, a3], [a1, a2, a3, s]]

After

derived = [[1, 2, 3, 4], [], [[[a1, t], 17], [[a2, t], 18], [[a3, t], 19], [[s, a0], 16]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S031

S033 · calculation

Before

MFMC cut mask15 OUTWARD capacity := sum(16, 17, 18, 19)

After

MFMC cut mask15 OUTWARD capacity = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · implies

Before

source roles = [15, [a0, a1, a2, a3], [a0, a1, a2, a3, s]]

After

derived = [[0, 1, 2, 3], [], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[a3, t], 19]], 70]

Conditions: none

S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.

Step identifiers and dependencies

Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S033

S035 · implies

Before

source roles = [[a0, a1, a2, a3, s, t], [a0, a1, a2, a3]]

After

derived = [16, 16, 70, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]]]

Conditions: none

Every s–t cut uniquely corresponds to a subset of other vertices,so all2^(n−2) cuts are compared.Select the minimum and all minimizers from this same complete family,retaining mask tie order.

Step identifiers and dependencies

Rule: MFMC_ALL_cut_family_complete_minimum_and_ALL_ties · Uses: G001, S034

S036 · calculation

Before

MFMC vertex a0 ALL incoming := sum(16)

After

MFMC vertex a0 ALL incoming = 16

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

MFMC vertex a0 ALL outgoing := sum(16)

After

MFMC vertex a0 ALL outgoing = 16

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

MFMC vertex a1 ALL incoming := sum(17)

After

MFMC vertex a1 ALL incoming = 17

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

MFMC vertex a1 ALL outgoing := sum(17)

After

MFMC vertex a1 ALL outgoing = 17

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

MFMC vertex a2 ALL incoming := sum(18)

After

MFMC vertex a2 ALL incoming = 18

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

MFMC vertex a2 ALL outgoing := sum(18)

After

MFMC vertex a2 ALL outgoing = 18

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

MFMC vertex a3 ALL incoming := sum(19)

After

MFMC vertex a3 ALL incoming = 19

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

MFMC vertex a3 ALL outgoing := sum(19)

After

MFMC vertex a3 ALL outgoing = 19

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

MFMC vertex s ALL incoming := sum()

After

MFMC vertex s ALL incoming = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

MFMC vertex s ALL outgoing := sum(16, 17, 18, 19)

After

MFMC vertex s ALL outgoing = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

MFMC vertex t ALL incoming := sum(16, 17, 18, 19)

After

MFMC vertex t ALL incoming = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

MFMC vertex t ALL outgoing := sum()

After

MFMC vertex t ALL outgoing = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

MFMC SOURCE net out not gross := sub(70, 0)

After

MFMC SOURCE net out not gross = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

MFMC SINK net in not gross := sub(70, 0)

After

MFMC SINK net in not gross = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · implies

Before

source roles = [[[a0, t], [a1, t], [a2, t], [a3, t], [s, a0], [s, a1], [s, a2], [s, a3]], [16, 17, 18, 19, 16, 17, 18, 19], [16, 17, 18, 19, 16, 17, 18, 19], s, t]

After

derived = [[[[a0, t], 16, 16, Yes], [[a1, t], 17, 17, Yes], [[a2, t], 18, 18, Yes], [[a3, t], 19, 19, Yes], [[s, a0], 16, 16, Yes], [[s, a1], 17, 17, Yes], [[s, a2], 18, 18, Yes], [[s, a3], 19, 19, Yes]], [[a0, 16, 16, Yes], [a1, 17, 17, Yes], [a2, 18, 18, Yes], [a3, 19, 19, Yes]], Yes, Yes, 70, 70, Yes, Yes]

Conditions: none

Feasibility requires every edge capacity bound,every internal conservation equation and equal source/sink NET values.Do not discard incoming-source or outgoing-sink arcs.A feasible flow may have negative net value;do not add value>=0.

Step identifiers and dependencies

Rule: MFMC_EVERY_capacity_EVERY_internal_balance_source_sink_NET_feasibility · Uses: G001, S049

S051 · calculation

Before

MFMC cut mask0 OUTWARD flow := sum(16, 17, 18, 19)

After

MFMC cut mask0 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

MFMC cut mask0 INWARD flow := sum()

After

MFMC cut mask0 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

MFMC cut mask0 NET flow := sub(70, 0)

After

MFMC cut mask0 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · implies

Before

source roles = [[s], [4, 5, 6, 7], [], 70]

After

derived = [[[s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S053

S055 · calculation

Before

MFMC cut mask1 OUTWARD flow := sum(16, 17, 18, 19)

After

MFMC cut mask1 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · calculation

Before

MFMC cut mask1 INWARD flow := sum()

After

MFMC cut mask1 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · calculation

Before

MFMC cut mask1 NET flow := sub(70, 0)

After

MFMC cut mask1 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · implies

Before

source roles = [[a0, s], [0, 5, 6, 7], [], 70]

After

derived = [[[a0, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S057

S059 · calculation

Before

MFMC cut mask2 OUTWARD flow := sum(17, 16, 18, 19)

After

MFMC cut mask2 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · calculation

Before

MFMC cut mask2 INWARD flow := sum()

After

MFMC cut mask2 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · calculation

Before

MFMC cut mask2 NET flow := sub(70, 0)

After

MFMC cut mask2 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · implies

Before

source roles = [[a1, s], [1, 4, 6, 7], [], 70]

After

derived = [[[a1, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S061

S063 · calculation

Before

MFMC cut mask3 OUTWARD flow := sum(16, 17, 18, 19)

After

MFMC cut mask3 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · calculation

Before

MFMC cut mask3 INWARD flow := sum()

After

MFMC cut mask3 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · calculation

Before

MFMC cut mask3 NET flow := sub(70, 0)

After

MFMC cut mask3 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · implies

Before

source roles = [[a0, a1, s], [0, 1, 6, 7], [], 70]

After

derived = [[[a0, a1, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S065

S067 · calculation

Before

MFMC cut mask4 OUTWARD flow := sum(18, 16, 17, 19)

After

MFMC cut mask4 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · calculation

Before

MFMC cut mask4 INWARD flow := sum()

After

MFMC cut mask4 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · calculation

Before

MFMC cut mask4 NET flow := sub(70, 0)

After

MFMC cut mask4 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · implies

Before

source roles = [[a2, s], [2, 4, 5, 7], [], 70]

After

derived = [[[a2, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S069

S071 · calculation

Before

MFMC cut mask5 OUTWARD flow := sum(16, 18, 17, 19)

After

MFMC cut mask5 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · calculation

Before

MFMC cut mask5 INWARD flow := sum()

After

MFMC cut mask5 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · calculation

Before

MFMC cut mask5 NET flow := sub(70, 0)

After

MFMC cut mask5 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · implies

Before

source roles = [[a0, a2, s], [0, 2, 5, 7], [], 70]

After

derived = [[[a0, a2, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S073

S075 · calculation

Before

MFMC cut mask6 OUTWARD flow := sum(17, 18, 16, 19)

After

MFMC cut mask6 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · calculation

Before

MFMC cut mask6 INWARD flow := sum()

After

MFMC cut mask6 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S075

S077 · calculation

Before

MFMC cut mask6 NET flow := sub(70, 0)

After

MFMC cut mask6 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S076

S078 · implies

Before

source roles = [[a1, a2, s], [1, 2, 4, 7], [], 70]

After

derived = [[[a1, a2, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S077

S079 · calculation

Before

MFMC cut mask7 OUTWARD flow := sum(16, 17, 18, 19)

After

MFMC cut mask7 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S078

S080 · calculation

Before

MFMC cut mask7 INWARD flow := sum()

After

MFMC cut mask7 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S079

S081 · calculation

Before

MFMC cut mask7 NET flow := sub(70, 0)

After

MFMC cut mask7 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S080

S082 · implies

Before

source roles = [[a0, a1, a2, s], [0, 1, 2, 7], [], 70]

After

derived = [[[a0, a1, a2, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S081

S083 · calculation

Before

MFMC cut mask8 OUTWARD flow := sum(19, 16, 17, 18)

After

MFMC cut mask8 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S082

S084 · calculation

Before

MFMC cut mask8 INWARD flow := sum()

After

MFMC cut mask8 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S083

S085 · calculation

Before

MFMC cut mask8 NET flow := sub(70, 0)

After

MFMC cut mask8 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S084

S086 · implies

Before

source roles = [[a3, s], [3, 4, 5, 6], [], 70]

After

derived = [[[a3, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S085

S087 · calculation

Before

MFMC cut mask9 OUTWARD flow := sum(16, 19, 17, 18)

After

MFMC cut mask9 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S086

S088 · calculation

Before

MFMC cut mask9 INWARD flow := sum()

After

MFMC cut mask9 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S087

S089 · calculation

Before

MFMC cut mask9 NET flow := sub(70, 0)

After

MFMC cut mask9 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S088

S090 · implies

Before

source roles = [[a0, a3, s], [0, 3, 5, 6], [], 70]

After

derived = [[[a0, a3, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S089

S091 · calculation

Before

MFMC cut mask10 OUTWARD flow := sum(17, 19, 16, 18)

After

MFMC cut mask10 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S090

S092 · calculation

Before

MFMC cut mask10 INWARD flow := sum()

After

MFMC cut mask10 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S091

S093 · calculation

Before

MFMC cut mask10 NET flow := sub(70, 0)

After

MFMC cut mask10 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S092

S094 · implies

Before

source roles = [[a1, a3, s], [1, 3, 4, 6], [], 70]

After

derived = [[[a1, a3, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S093

S095 · calculation

Before

MFMC cut mask11 OUTWARD flow := sum(16, 17, 19, 18)

After

MFMC cut mask11 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S094

S096 · calculation

Before

MFMC cut mask11 INWARD flow := sum()

After

MFMC cut mask11 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S095

S097 · calculation

Before

MFMC cut mask11 NET flow := sub(70, 0)

After

MFMC cut mask11 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S096

S098 · implies

Before

source roles = [[a0, a1, a3, s], [0, 1, 3, 6], [], 70]

After

derived = [[[a0, a1, a3, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S097

S099 · calculation

Before

MFMC cut mask12 OUTWARD flow := sum(18, 19, 16, 17)

After

MFMC cut mask12 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S098

S100 · calculation

Before

MFMC cut mask12 INWARD flow := sum()

After

MFMC cut mask12 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S099

S101 · calculation

Before

MFMC cut mask12 NET flow := sub(70, 0)

After

MFMC cut mask12 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S100

S102 · implies

Before

source roles = [[a2, a3, s], [2, 3, 4, 5], [], 70]

After

derived = [[[a2, a3, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S101

S103 · calculation

Before

MFMC cut mask13 OUTWARD flow := sum(16, 18, 19, 17)

After

MFMC cut mask13 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S102

S104 · calculation

Before

MFMC cut mask13 INWARD flow := sum()

After

MFMC cut mask13 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S103

S105 · calculation

Before

MFMC cut mask13 NET flow := sub(70, 0)

After

MFMC cut mask13 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S104

S106 · implies

Before

source roles = [[a0, a2, a3, s], [0, 2, 3, 5], [], 70]

After

derived = [[[a0, a2, a3, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S105

S107 · calculation

Before

MFMC cut mask14 OUTWARD flow := sum(17, 18, 19, 16)

After

MFMC cut mask14 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S106

S108 · calculation

Before

MFMC cut mask14 INWARD flow := sum()

After

MFMC cut mask14 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S107

S109 · calculation

Before

MFMC cut mask14 NET flow := sub(70, 0)

After

MFMC cut mask14 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S108

S110 · implies

Before

source roles = [[a1, a2, a3, s], [1, 2, 3, 4], [], 70]

After

derived = [[[a1, a2, a3, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S109

S111 · calculation

Before

MFMC cut mask15 OUTWARD flow := sum(16, 17, 18, 19)

After

MFMC cut mask15 OUTWARD flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S110

S112 · calculation

Before

MFMC cut mask15 INWARD flow := sum()

After

MFMC cut mask15 INWARD flow = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S111

S113 · calculation

Before

MFMC cut mask15 NET flow := sub(70, 0)

After

MFMC cut mask15 NET flow = 70

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S112

S114 · implies

Before

source roles = [[a0, a1, a2, a3, s], [0, 1, 2, 3], [], 70]

After

derived = [[[a0, a1, a2, a3, s], 70, 0, 70, 70, Yes, Yes]]

Conditions: none

Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.

Step identifiers and dependencies

Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S113

S115 · implies

Before

source roles = [Yes, 70, 70, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]]]

After

derived = [[[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No]

Conditions: none

Any feasible flow is bounded by this equal-valued cut,and any cut is bounded below by this feasible flow,so the current flow is maximum and equal-valued cuts minimum.This uses universal weak bounds plus feasibility,not bare equality.The final No only withholds integrality/algorithm proof.

Step identifiers and dependencies

Rule: MFMC_feasible_equal_value_GENERAL_bounds_certify_BOTH_optima · Uses: G001, S114

Reverse checks

  • S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S113 · premise_and_result_audit — Recompute MFMC cut mask15 NET flow with exact rational arithmetic, not rounded decimals.
  • S112 · premise_and_result_audit — Recompute MFMC cut mask15 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S111 · premise_and_result_audit — Recompute MFMC cut mask15 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S109 · premise_and_result_audit — Recompute MFMC cut mask14 NET flow with exact rational arithmetic, not rounded decimals.
  • S108 · premise_and_result_audit — Recompute MFMC cut mask14 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S107 · premise_and_result_audit — Recompute MFMC cut mask14 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S105 · premise_and_result_audit — Recompute MFMC cut mask13 NET flow with exact rational arithmetic, not rounded decimals.
  • S104 · premise_and_result_audit — Recompute MFMC cut mask13 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S103 · premise_and_result_audit — Recompute MFMC cut mask13 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S101 · premise_and_result_audit — Recompute MFMC cut mask12 NET flow with exact rational arithmetic, not rounded decimals.
  • S100 · premise_and_result_audit — Recompute MFMC cut mask12 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S099 · premise_and_result_audit — Recompute MFMC cut mask12 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S097 · premise_and_result_audit — Recompute MFMC cut mask11 NET flow with exact rational arithmetic, not rounded decimals.
  • S096 · premise_and_result_audit — Recompute MFMC cut mask11 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S095 · premise_and_result_audit — Recompute MFMC cut mask11 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S093 · premise_and_result_audit — Recompute MFMC cut mask10 NET flow with exact rational arithmetic, not rounded decimals.
  • S092 · premise_and_result_audit — Recompute MFMC cut mask10 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S091 · premise_and_result_audit — Recompute MFMC cut mask10 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S089 · premise_and_result_audit — Recompute MFMC cut mask9 NET flow with exact rational arithmetic, not rounded decimals.
  • S088 · premise_and_result_audit — Recompute MFMC cut mask9 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S087 · premise_and_result_audit — Recompute MFMC cut mask9 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S085 · premise_and_result_audit — Recompute MFMC cut mask8 NET flow with exact rational arithmetic, not rounded decimals.
  • S084 · premise_and_result_audit — Recompute MFMC cut mask8 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S083 · premise_and_result_audit — Recompute MFMC cut mask8 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S081 · premise_and_result_audit — Recompute MFMC cut mask7 NET flow with exact rational arithmetic, not rounded decimals.
  • S080 · premise_and_result_audit — Recompute MFMC cut mask7 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S079 · premise_and_result_audit — Recompute MFMC cut mask7 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S077 · premise_and_result_audit — Recompute MFMC cut mask6 NET flow with exact rational arithmetic, not rounded decimals.
  • S076 · premise_and_result_audit — Recompute MFMC cut mask6 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S075 · premise_and_result_audit — Recompute MFMC cut mask6 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S073 · premise_and_result_audit — Recompute MFMC cut mask5 NET flow with exact rational arithmetic, not rounded decimals.
  • S072 · premise_and_result_audit — Recompute MFMC cut mask5 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S071 · premise_and_result_audit — Recompute MFMC cut mask5 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S069 · premise_and_result_audit — Recompute MFMC cut mask4 NET flow with exact rational arithmetic, not rounded decimals.
  • S068 · premise_and_result_audit — Recompute MFMC cut mask4 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S067 · premise_and_result_audit — Recompute MFMC cut mask4 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S065 · premise_and_result_audit — Recompute MFMC cut mask3 NET flow with exact rational arithmetic, not rounded decimals.
  • S064 · premise_and_result_audit — Recompute MFMC cut mask3 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S063 · premise_and_result_audit — Recompute MFMC cut mask3 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S061 · premise_and_result_audit — Recompute MFMC cut mask2 NET flow with exact rational arithmetic, not rounded decimals.
  • S060 · premise_and_result_audit — Recompute MFMC cut mask2 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S059 · premise_and_result_audit — Recompute MFMC cut mask2 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S057 · premise_and_result_audit — Recompute MFMC cut mask1 NET flow with exact rational arithmetic, not rounded decimals.
  • S056 · premise_and_result_audit — Recompute MFMC cut mask1 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S055 · premise_and_result_audit — Recompute MFMC cut mask1 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S053 · premise_and_result_audit — Recompute MFMC cut mask0 NET flow with exact rational arithmetic, not rounded decimals.
  • S052 · premise_and_result_audit — Recompute MFMC cut mask0 INWARD flow with exact rational arithmetic, not rounded decimals.
  • S051 · premise_and_result_audit — Recompute MFMC cut mask0 OUTWARD flow with exact rational arithmetic, not rounded decimals.
  • S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S049 · premise_and_result_audit — Recompute MFMC SINK net in not gross with exact rational arithmetic, not rounded decimals.
  • S048 · premise_and_result_audit — Recompute MFMC SOURCE net out not gross with exact rational arithmetic, not rounded decimals.
  • S047 · premise_and_result_audit — Recompute MFMC vertex t ALL outgoing with exact rational arithmetic, not rounded decimals.
  • S046 · premise_and_result_audit — Recompute MFMC vertex t ALL incoming with exact rational arithmetic, not rounded decimals.
  • S045 · premise_and_result_audit — Recompute MFMC vertex s ALL outgoing with exact rational arithmetic, not rounded decimals.
  • S044 · premise_and_result_audit — Recompute MFMC vertex s ALL incoming with exact rational arithmetic, not rounded decimals.
  • S043 · premise_and_result_audit — Recompute MFMC vertex a3 ALL outgoing with exact rational arithmetic, not rounded decimals.
  • S042 · premise_and_result_audit — Recompute MFMC vertex a3 ALL incoming with exact rational arithmetic, not rounded decimals.
  • S041 · premise_and_result_audit — Recompute MFMC vertex a2 ALL outgoing with exact rational arithmetic, not rounded decimals.
  • S040 · premise_and_result_audit — Recompute MFMC vertex a2 ALL incoming with exact rational arithmetic, not rounded decimals.
  • S039 · premise_and_result_audit — Recompute MFMC vertex a1 ALL outgoing with exact rational arithmetic, not rounded decimals.
  • S038 · premise_and_result_audit — Recompute MFMC vertex a1 ALL incoming with exact rational arithmetic, not rounded decimals.
  • S037 · premise_and_result_audit — Recompute MFMC vertex a0 ALL outgoing with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Recompute MFMC vertex a0 ALL incoming with exact rational arithmetic, not rounded decimals.
  • S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S033 · premise_and_result_audit — Recompute MFMC cut mask15 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Recompute MFMC cut mask14 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Recompute MFMC cut mask13 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Recompute MFMC cut mask12 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Recompute MFMC cut mask11 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Recompute MFMC cut mask10 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Recompute MFMC cut mask9 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Recompute MFMC cut mask8 OUTWARD capacity 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 MFMC cut mask7 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Recompute MFMC cut mask6 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Recompute MFMC cut mask5 OUTWARD capacity 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 MFMC cut mask4 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Recompute MFMC cut mask3 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Recompute MFMC cut mask2 OUTWARD capacity 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 — Recompute MFMC cut mask1 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
  • 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 — Recompute MFMC cut mask0 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
  • 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

(70, 70, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

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: ffc185449614cc1aba76511c4f4ec6bd49f29d3b3b09bae5055d6cd0ca3985c6
  • candidate record SHA-256: 38cdb7406eb871c0ba52ad87e82472002a72a81f437aa2bf95945c0fcd8ad91d
  • semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82
  • question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON