Question 25953: A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

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=[4, 4, 5, 5, 6, 6, 7, 7], candidate flows=[3, 3, 4, 4, 5, 5, 6, 6]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_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=[4, 4, 5, 5, 6, 6, 7, 7], candidate flows=[3, 3, 4, 4, 5, 5, 6, 6]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_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

Step-by-step derivation

source roles = [mfmc-flow-cut-upper-bound]

derived = [[source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate]]

source roles = [[s, a0, a1, a2, a3, t], [[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], [4, 4, 5, 5, 6, 6, 7, 7], [3, 3, 4, 4, 5, 5, 6, 6], s, t]

derived = [[a0, a1, a2, a3, s, t], [[a0, t], [a1, t], [a2, t], [a3, t], [s, a0], [s, a1], [s, a2], [s, a3]], [4, 5, 6, 7, 4, 5, 6, 7], [3, 4, 5, 6, 3, 4, 5, 6]]

MFMC cut mask0 OUTWARD capacity := sum(4, 5, 6, 7)

MFMC cut mask0 OUTWARD capacity = 22

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

derived = [[4, 5, 6, 7], [], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22]

MFMC cut mask1 OUTWARD capacity := sum(4, 5, 6, 7)

MFMC cut mask1 OUTWARD capacity = 22

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

derived = [[0, 5, 6, 7], [], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22]

MFMC cut mask2 OUTWARD capacity := sum(5, 4, 6, 7)

MFMC cut mask2 OUTWARD capacity = 22

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

derived = [[1, 4, 6, 7], [], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22]

MFMC cut mask3 OUTWARD capacity := sum(4, 5, 6, 7)

MFMC cut mask3 OUTWARD capacity = 22

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

derived = [[0, 1, 6, 7], [], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22]

MFMC cut mask4 OUTWARD capacity := sum(6, 4, 5, 7)

MFMC cut mask4 OUTWARD capacity = 22

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

derived = [[2, 4, 5, 7], [], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22]

MFMC cut mask5 OUTWARD capacity := sum(4, 6, 5, 7)

MFMC cut mask5 OUTWARD capacity = 22

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

derived = [[0, 2, 5, 7], [], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22]

MFMC cut mask6 OUTWARD capacity := sum(5, 6, 4, 7)

MFMC cut mask6 OUTWARD capacity = 22

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

derived = [[1, 2, 4, 7], [], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22]

MFMC cut mask7 OUTWARD capacity := sum(4, 5, 6, 7)

MFMC cut mask7 OUTWARD capacity = 22

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

derived = [[0, 1, 2, 7], [], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22]

MFMC cut mask8 OUTWARD capacity := sum(7, 4, 5, 6)

MFMC cut mask8 OUTWARD capacity = 22

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

derived = [[3, 4, 5, 6], [], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22]

MFMC cut mask9 OUTWARD capacity := sum(4, 7, 5, 6)

MFMC cut mask9 OUTWARD capacity = 22

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

derived = [[0, 3, 5, 6], [], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22]

MFMC cut mask10 OUTWARD capacity := sum(5, 7, 4, 6)

MFMC cut mask10 OUTWARD capacity = 22

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

derived = [[1, 3, 4, 6], [], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22]

MFMC cut mask11 OUTWARD capacity := sum(4, 5, 7, 6)

MFMC cut mask11 OUTWARD capacity = 22

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

derived = [[0, 1, 3, 6], [], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22]

MFMC cut mask12 OUTWARD capacity := sum(6, 7, 4, 5)

MFMC cut mask12 OUTWARD capacity = 22

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

derived = [[2, 3, 4, 5], [], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22]

MFMC cut mask13 OUTWARD capacity := sum(4, 6, 7, 5)

MFMC cut mask13 OUTWARD capacity = 22

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

derived = [[0, 2, 3, 5], [], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22]

MFMC cut mask14 OUTWARD capacity := sum(5, 6, 7, 4)

MFMC cut mask14 OUTWARD capacity = 22

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

derived = [[1, 2, 3, 4], [], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22]

MFMC cut mask15 OUTWARD capacity := sum(4, 5, 6, 7)

MFMC cut mask15 OUTWARD capacity = 22

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

derived = [[0, 1, 2, 3], [], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]

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

derived = [16, 16, 22, [[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]]]

MFMC vertex a0 ALL incoming := sum(3)

MFMC vertex a0 ALL incoming = 3

MFMC vertex a0 ALL outgoing := sum(3)

MFMC vertex a0 ALL outgoing = 3

MFMC vertex a1 ALL incoming := sum(4)

MFMC vertex a1 ALL incoming = 4

MFMC vertex a1 ALL outgoing := sum(4)

MFMC vertex a1 ALL outgoing = 4

MFMC vertex a2 ALL incoming := sum(5)

MFMC vertex a2 ALL incoming = 5

MFMC vertex a2 ALL outgoing := sum(5)

MFMC vertex a2 ALL outgoing = 5

MFMC vertex a3 ALL incoming := sum(6)

MFMC vertex a3 ALL incoming = 6

MFMC vertex a3 ALL outgoing := sum(6)

MFMC vertex a3 ALL outgoing = 6

MFMC vertex s ALL incoming := sum()

MFMC vertex s ALL incoming = 0

MFMC vertex s ALL outgoing := sum(3, 4, 5, 6)

MFMC vertex s ALL outgoing = 18

MFMC vertex t ALL incoming := sum(3, 4, 5, 6)

MFMC vertex t ALL incoming = 18

MFMC vertex t ALL outgoing := sum()

MFMC vertex t ALL outgoing = 0

MFMC SOURCE net out not gross := sub(18, 0)

MFMC SOURCE net out not gross = 18

MFMC SINK net in not gross := sub(18, 0)

MFMC SINK net in not gross = 18

source roles = [[[a0, t], [a1, t], [a2, t], [a3, t], [s, a0], [s, a1], [s, a2], [s, a3]], [3, 4, 5, 6, 3, 4, 5, 6], [4, 5, 6, 7, 4, 5, 6, 7], s, t]

derived = [[[[a0, t], 3, 4, Yes], [[a1, t], 4, 5, Yes], [[a2, t], 5, 6, Yes], [[a3, t], 6, 7, Yes], [[s, a0], 3, 4, Yes], [[s, a1], 4, 5, Yes], [[s, a2], 5, 6, Yes], [[s, a3], 6, 7, Yes]], [[a0, 3, 3, Yes], [a1, 4, 4, Yes], [a2, 5, 5, Yes], [a3, 6, 6, Yes]], Yes, Yes, 18, 18, Yes, Yes]

MFMC cut mask0 OUTWARD flow := sum(3, 4, 5, 6)

MFMC cut mask0 OUTWARD flow = 18

MFMC cut mask0 INWARD flow := sum()

MFMC cut mask0 INWARD flow = 0

MFMC cut mask0 NET flow := sub(18, 0)

MFMC cut mask0 NET flow = 18

source roles = [[s], [4, 5, 6, 7], [], 18]

derived = [[[s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask1 OUTWARD flow := sum(3, 4, 5, 6)

MFMC cut mask1 OUTWARD flow = 18

MFMC cut mask1 INWARD flow := sum()

MFMC cut mask1 INWARD flow = 0

MFMC cut mask1 NET flow := sub(18, 0)

MFMC cut mask1 NET flow = 18

source roles = [[a0, s], [0, 5, 6, 7], [], 18]

derived = [[[a0, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask2 OUTWARD flow := sum(4, 3, 5, 6)

MFMC cut mask2 OUTWARD flow = 18

MFMC cut mask2 INWARD flow := sum()

MFMC cut mask2 INWARD flow = 0

MFMC cut mask2 NET flow := sub(18, 0)

MFMC cut mask2 NET flow = 18

source roles = [[a1, s], [1, 4, 6, 7], [], 18]

derived = [[[a1, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask3 OUTWARD flow := sum(3, 4, 5, 6)

MFMC cut mask3 OUTWARD flow = 18

MFMC cut mask3 INWARD flow := sum()

MFMC cut mask3 INWARD flow = 0

MFMC cut mask3 NET flow := sub(18, 0)

MFMC cut mask3 NET flow = 18

source roles = [[a0, a1, s], [0, 1, 6, 7], [], 18]

derived = [[[a0, a1, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask4 OUTWARD flow := sum(5, 3, 4, 6)

MFMC cut mask4 OUTWARD flow = 18

MFMC cut mask4 INWARD flow := sum()

MFMC cut mask4 INWARD flow = 0

MFMC cut mask4 NET flow := sub(18, 0)

MFMC cut mask4 NET flow = 18

source roles = [[a2, s], [2, 4, 5, 7], [], 18]

derived = [[[a2, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask5 OUTWARD flow := sum(3, 5, 4, 6)

MFMC cut mask5 OUTWARD flow = 18

MFMC cut mask5 INWARD flow := sum()

MFMC cut mask5 INWARD flow = 0

MFMC cut mask5 NET flow := sub(18, 0)

MFMC cut mask5 NET flow = 18

source roles = [[a0, a2, s], [0, 2, 5, 7], [], 18]

derived = [[[a0, a2, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask6 OUTWARD flow := sum(4, 5, 3, 6)

MFMC cut mask6 OUTWARD flow = 18

MFMC cut mask6 INWARD flow := sum()

MFMC cut mask6 INWARD flow = 0

MFMC cut mask6 NET flow := sub(18, 0)

MFMC cut mask6 NET flow = 18

source roles = [[a1, a2, s], [1, 2, 4, 7], [], 18]

derived = [[[a1, a2, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask7 OUTWARD flow := sum(3, 4, 5, 6)

MFMC cut mask7 OUTWARD flow = 18

MFMC cut mask7 INWARD flow := sum()

MFMC cut mask7 INWARD flow = 0

MFMC cut mask7 NET flow := sub(18, 0)

MFMC cut mask7 NET flow = 18

source roles = [[a0, a1, a2, s], [0, 1, 2, 7], [], 18]

derived = [[[a0, a1, a2, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask8 OUTWARD flow := sum(6, 3, 4, 5)

MFMC cut mask8 OUTWARD flow = 18

MFMC cut mask8 INWARD flow := sum()

MFMC cut mask8 INWARD flow = 0

MFMC cut mask8 NET flow := sub(18, 0)

MFMC cut mask8 NET flow = 18

source roles = [[a3, s], [3, 4, 5, 6], [], 18]

derived = [[[a3, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask9 OUTWARD flow := sum(3, 6, 4, 5)

MFMC cut mask9 OUTWARD flow = 18

MFMC cut mask9 INWARD flow := sum()

MFMC cut mask9 INWARD flow = 0

MFMC cut mask9 NET flow := sub(18, 0)

MFMC cut mask9 NET flow = 18

source roles = [[a0, a3, s], [0, 3, 5, 6], [], 18]

derived = [[[a0, a3, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask10 OUTWARD flow := sum(4, 6, 3, 5)

MFMC cut mask10 OUTWARD flow = 18

MFMC cut mask10 INWARD flow := sum()

MFMC cut mask10 INWARD flow = 0

MFMC cut mask10 NET flow := sub(18, 0)

MFMC cut mask10 NET flow = 18

source roles = [[a1, a3, s], [1, 3, 4, 6], [], 18]

derived = [[[a1, a3, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask11 OUTWARD flow := sum(3, 4, 6, 5)

MFMC cut mask11 OUTWARD flow = 18

MFMC cut mask11 INWARD flow := sum()

MFMC cut mask11 INWARD flow = 0

MFMC cut mask11 NET flow := sub(18, 0)

MFMC cut mask11 NET flow = 18

source roles = [[a0, a1, a3, s], [0, 1, 3, 6], [], 18]

derived = [[[a0, a1, a3, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask12 OUTWARD flow := sum(5, 6, 3, 4)

MFMC cut mask12 OUTWARD flow = 18

MFMC cut mask12 INWARD flow := sum()

MFMC cut mask12 INWARD flow = 0

MFMC cut mask12 NET flow := sub(18, 0)

MFMC cut mask12 NET flow = 18

source roles = [[a2, a3, s], [2, 3, 4, 5], [], 18]

derived = [[[a2, a3, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask13 OUTWARD flow := sum(3, 5, 6, 4)

MFMC cut mask13 OUTWARD flow = 18

MFMC cut mask13 INWARD flow := sum()

MFMC cut mask13 INWARD flow = 0

MFMC cut mask13 NET flow := sub(18, 0)

MFMC cut mask13 NET flow = 18

source roles = [[a0, a2, a3, s], [0, 2, 3, 5], [], 18]

derived = [[[a0, a2, a3, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask14 OUTWARD flow := sum(4, 5, 6, 3)

MFMC cut mask14 OUTWARD flow = 18

MFMC cut mask14 INWARD flow := sum()

MFMC cut mask14 INWARD flow = 0

MFMC cut mask14 NET flow := sub(18, 0)

MFMC cut mask14 NET flow = 18

source roles = [[a1, a2, a3, s], [1, 2, 3, 4], [], 18]

derived = [[[a1, a2, a3, s], 18, 0, 18, 22, Yes, Yes]]

MFMC cut mask15 OUTWARD flow := sum(3, 4, 5, 6)

MFMC cut mask15 OUTWARD flow = 18

MFMC cut mask15 INWARD flow := sum()

MFMC cut mask15 INWARD flow = 0

MFMC cut mask15 NET flow := sub(18, 0)

MFMC cut mask15 NET flow = 18

source roles = [[a0, a1, a2, a3, s], [0, 1, 2, 3], [], 18]

derived = [[[a0, a1, a2, a3, s], 18, 0, 18, 22, Yes, Yes]]

Reverse checks

• 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

(18, 18, [[[s], 18, 0, 18, 22, Yes, Yes], [[a0, s], 18, 0, 18, 22, Yes, Yes], [[a1, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, s], 18, 0, 18, 22, Yes, Yes], [[a2, s], 18, 0, 18, 22, Yes, Yes], [[a0, a2, s], 18, 0, 18, 22, Yes, Yes], [[a1, a2, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a2, s], 18, 0, 18, 22, Yes, Yes], [[a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a3, s], 18, 0, 18, 22, Yes, Yes], [[a1, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a3, s], 18, 0, 18, 22, Yes, Yes], [[a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a1, a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a2, a3, s], 18, 0, 18, 22, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

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.