# Question 25930: An Equal-Valued Feasible Flow and Cut Certify Both Optima

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0050-25930 · q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0080



## Original question and givens

Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[6, 6, 7, 7], candidate flows=[6, 6, 7, 7]. 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, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[6, 6, 7, 7], candidate flows=[6, 6, 7, 7]. 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.


- [mko-max-flow-min-cut-theorem](https://ocme.evemisslab.com/en/mko/mko-max-flow-min-cut-theorem/)





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [[s, a0, a1, t], [[s, a0], [a0, t], [s, a1], [a1, t]], [6, 6, 7, 7], [6, 6, 7, 7], s, t]

After

derived = [[a0, a1, s, t], [[a0, t], [a1, t], [s, a0], [s, a1]], [6, 7, 6, 7], [6, 7, 6, 7]]

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 dependenciesRule: MFMC_complete_directed_network_ZIP_before_source_order · Uses: G001, S001





S003 · calculationBefore

MFMC cut mask0 OUTWARD capacity := sum(6, 7)

After

MFMC cut mask0 OUTWARD capacity = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

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

After

derived = [[2, 3], [], [[[s, a0], 6], [[s, a1], 7]], 13]

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 dependenciesRule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S003





S005 · calculationBefore

MFMC cut mask1 OUTWARD capacity := sum(6, 7)

After

MFMC cut mask1 OUTWARD capacity = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

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

After

derived = [[0, 3], [], [[[a0, t], 6], [[s, a1], 7]], 13]

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 dependenciesRule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S005





S007 · calculationBefore

MFMC cut mask2 OUTWARD capacity := sum(7, 6)

After

MFMC cut mask2 OUTWARD capacity = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

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

After

derived = [[1, 2], [], [[[a1, t], 7], [[s, a0], 6]], 13]

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 dependenciesRule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S007





S009 · calculationBefore

MFMC cut mask3 OUTWARD capacity := sum(6, 7)

After

MFMC cut mask3 OUTWARD capacity = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

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

After

derived = [[0, 1], [], [[[a0, t], 6], [[a1, t], 7]], 13]

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 dependenciesRule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S009





S011 · impliesBefore

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

After

derived = [4, 4, 13, [[s], [a0, s], [a1, s], [a0, a1, 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 dependenciesRule: MFMC_ALL_cut_family_complete_minimum_and_ALL_ties · Uses: G001, S010





S012 · calculationBefore

MFMC vertex a0 ALL incoming := sum(6)

After

MFMC vertex a0 ALL incoming = 6

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

MFMC vertex a0 ALL outgoing := sum(6)

After

MFMC vertex a0 ALL outgoing = 6

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

MFMC vertex a1 ALL incoming := sum(7)

After

MFMC vertex a1 ALL incoming = 7

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · calculationBefore

MFMC vertex a1 ALL outgoing := sum(7)

After

MFMC vertex a1 ALL outgoing = 7

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

MFMC vertex s ALL outgoing := sum(6, 7)

After

MFMC vertex s ALL outgoing = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · calculationBefore

MFMC vertex t ALL incoming := sum(6, 7)

After

MFMC vertex t ALL incoming = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · calculationBefore

MFMC SOURCE net out not gross := sub(13, 0)

After

MFMC SOURCE net out not gross = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · calculationBefore

MFMC SINK net in not gross := sub(13, 0)

After

MFMC SINK net in not gross = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · impliesBefore

source roles = [[[a0, t], [a1, t], [s, a0], [s, a1]], [6, 7, 6, 7], [6, 7, 6, 7], s, t]

After

derived = [[[[a0, t], 6, 6, Yes], [[a1, t], 7, 7, Yes], [[s, a0], 6, 6, Yes], [[s, a1], 7, 7, Yes]], [[a0, 6, 6, Yes], [a1, 7, 7, Yes]], Yes, Yes, 13, 13, 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 dependenciesRule: MFMC_EVERY_capacity_EVERY_internal_balance_source_sink_NET_feasibility · Uses: G001, S021





S023 · calculationBefore

MFMC cut mask0 OUTWARD flow := sum(6, 7)

After

MFMC cut mask0 OUTWARD flow = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S023





S025 · calculationBefore

MFMC cut mask0 NET flow := sub(13, 0)

After

MFMC cut mask0 NET flow = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · impliesBefore

source roles = [[s], [2, 3], [], 13]

After

derived = [[[s], 13, 0, 13, 13, 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 dependenciesRule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S025





S027 · calculationBefore

MFMC cut mask1 OUTWARD flow := sum(6, 7)

After

MFMC cut mask1 OUTWARD flow = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026





S028 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S027





S029 · calculationBefore

MFMC cut mask1 NET flow := sub(13, 0)

After

MFMC cut mask1 NET flow = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · impliesBefore

source roles = [[a0, s], [0, 3], [], 13]

After

derived = [[[a0, s], 13, 0, 13, 13, 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 dependenciesRule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S029





S031 · calculationBefore

MFMC cut mask2 OUTWARD flow := sum(7, 6)

After

MFMC cut mask2 OUTWARD flow = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S031





S033 · calculationBefore

MFMC cut mask2 NET flow := sub(13, 0)

After

MFMC cut mask2 NET flow = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032





S034 · impliesBefore

source roles = [[a1, s], [1, 2], [], 13]

After

derived = [[[a1, s], 13, 0, 13, 13, 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 dependenciesRule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S033





S035 · calculationBefore

MFMC cut mask3 OUTWARD flow := sum(6, 7)

After

MFMC cut mask3 OUTWARD flow = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S035





S037 · calculationBefore

MFMC cut mask3 NET flow := sub(13, 0)

After

MFMC cut mask3 NET flow = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036





S038 · impliesBefore

source roles = [[a0, a1, s], [0, 1], [], 13]

After

derived = [[[a0, a1, s], 13, 0, 13, 13, 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 dependenciesRule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S037





S039 · impliesBefore

source roles = [Yes, 13, 13, [[s], [a0, s], [a1, s], [a0, a1, s]]]

After

derived = [[[s], [a0, s], [a1, s], [a0, a1, 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 dependenciesRule: MFMC_feasible_equal_value_GENERAL_bounds_certify_BOTH_optima · Uses: G001, S038









## Reverse checks


- S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S037 · premise_and_result_audit — Recompute MFMC cut mask3 NET flow with exact rational arithmetic, not rounded decimals.
- S036 · premise_and_result_audit — Recompute MFMC cut mask3 INWARD flow with exact rational arithmetic, not rounded decimals.
- S035 · premise_and_result_audit — Recompute MFMC cut mask3 OUTWARD flow with exact rational arithmetic, not rounded decimals.
- 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 mask2 NET flow with exact rational arithmetic, not rounded decimals.
- S032 · premise_and_result_audit — Recompute MFMC cut mask2 INWARD flow with exact rational arithmetic, not rounded decimals.
- S031 · premise_and_result_audit — Recompute MFMC cut mask2 OUTWARD flow 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 mask1 NET flow with exact rational arithmetic, not rounded decimals.
- S028 · premise_and_result_audit — Recompute MFMC cut mask1 INWARD flow with exact rational arithmetic, not rounded decimals.
- S027 · premise_and_result_audit — Recompute MFMC cut mask1 OUTWARD flow 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 mask0 NET flow with exact rational arithmetic, not rounded decimals.
- S024 · premise_and_result_audit — Recompute MFMC cut mask0 INWARD flow with exact rational arithmetic, not rounded decimals.
- S023 · premise_and_result_audit — Recompute MFMC cut mask0 OUTWARD flow 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 SINK net in not gross with exact rational arithmetic, not rounded decimals.
- S020 · premise_and_result_audit — Recompute MFMC SOURCE net out not gross with exact rational arithmetic, not rounded decimals.
- S019 · premise_and_result_audit — Recompute MFMC vertex t ALL outgoing with exact rational arithmetic, not rounded decimals.
- S018 · premise_and_result_audit — Recompute MFMC vertex t ALL incoming with exact rational arithmetic, not rounded decimals.
- S017 · premise_and_result_audit — Recompute MFMC vertex s ALL outgoing with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Recompute MFMC vertex s ALL incoming with exact rational arithmetic, not rounded decimals.
- S015 · premise_and_result_audit — Recompute MFMC vertex a1 ALL outgoing with exact rational arithmetic, not rounded decimals.
- S014 · premise_and_result_audit — Recompute MFMC vertex a1 ALL incoming with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Recompute MFMC vertex a0 ALL outgoing with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute MFMC vertex a0 ALL incoming with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

(13, 13, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, 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: d4e64dea0b658a8eafe671f771ffa2e243418a047b2dcb4f9fb8aa3c92f1f16c
- candidate record SHA-256: cf30b832c6a265d0dd86577502340a69161c6b161b9f087c1ef580681926d05a
- semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82
- question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f
- Lean: NOT_RUN · human review: NOT_PERFORMED

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