Question 25969: 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], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[20, 20, 21, 21, 22, 22, 23, 23, 25], candidate flows=[19, 19, 20, 20, 21, 21, 22, 22, 0]. 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], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[20, 20, 21, 21, 22, 22, 23, 23, 25], candidate flows=[19, 19, 20, 20, 21, 21, 22, 22, 0]. 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], [a0, s]], [20, 20, 21, 21, 22, 22, 23, 23, 25], [19, 19, 20, 20, 21, 21, 22, 22, 0], s, t]

derived = [[a0, a1, a2, a3, s, t], [[a0, s], [a0, t], [a1, t], [a2, t], [a3, t], [s, a0], [s, a1], [s, a2], [s, a3]], [25, 20, 21, 22, 23, 20, 21, 22, 23], [0, 19, 20, 21, 22, 19, 20, 21, 22]]

MFMC cut mask0 OUTWARD capacity := sum(20, 21, 22, 23)

MFMC cut mask0 OUTWARD capacity = 86

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

derived = [[5, 6, 7, 8], [0], [[[s, a0], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86]

MFMC cut mask1 OUTWARD capacity := sum(20, 21, 22, 23)

MFMC cut mask1 OUTWARD capacity = 86

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

derived = [[1, 6, 7, 8], [], [[[a0, t], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86]

MFMC cut mask2 OUTWARD capacity := sum(21, 20, 22, 23)

MFMC cut mask2 OUTWARD capacity = 86

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

derived = [[2, 5, 7, 8], [0], [[[a1, t], 21], [[s, a0], 20], [[s, a2], 22], [[s, a3], 23]], 86]

MFMC cut mask3 OUTWARD capacity := sum(20, 21, 22, 23)

MFMC cut mask3 OUTWARD capacity = 86

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

derived = [[1, 2, 7, 8], [], [[[a0, t], 20], [[a1, t], 21], [[s, a2], 22], [[s, a3], 23]], 86]

MFMC cut mask4 OUTWARD capacity := sum(22, 20, 21, 23)

MFMC cut mask4 OUTWARD capacity = 86

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

derived = [[3, 5, 6, 8], [0], [[[a2, t], 22], [[s, a0], 20], [[s, a1], 21], [[s, a3], 23]], 86]

MFMC cut mask5 OUTWARD capacity := sum(20, 22, 21, 23)

MFMC cut mask5 OUTWARD capacity = 86

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

derived = [[1, 3, 6, 8], [], [[[a0, t], 20], [[a2, t], 22], [[s, a1], 21], [[s, a3], 23]], 86]

MFMC cut mask6 OUTWARD capacity := sum(21, 22, 20, 23)

MFMC cut mask6 OUTWARD capacity = 86

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

derived = [[2, 3, 5, 8], [0], [[[a1, t], 21], [[a2, t], 22], [[s, a0], 20], [[s, a3], 23]], 86]

MFMC cut mask7 OUTWARD capacity := sum(20, 21, 22, 23)

MFMC cut mask7 OUTWARD capacity = 86

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

derived = [[1, 2, 3, 8], [], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[s, a3], 23]], 86]

MFMC cut mask8 OUTWARD capacity := sum(23, 20, 21, 22)

MFMC cut mask8 OUTWARD capacity = 86

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

derived = [[4, 5, 6, 7], [0], [[[a3, t], 23], [[s, a0], 20], [[s, a1], 21], [[s, a2], 22]], 86]

MFMC cut mask9 OUTWARD capacity := sum(20, 23, 21, 22)

MFMC cut mask9 OUTWARD capacity = 86

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

derived = [[1, 4, 6, 7], [], [[[a0, t], 20], [[a3, t], 23], [[s, a1], 21], [[s, a2], 22]], 86]

MFMC cut mask10 OUTWARD capacity := sum(21, 23, 20, 22)

MFMC cut mask10 OUTWARD capacity = 86

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

derived = [[2, 4, 5, 7], [0], [[[a1, t], 21], [[a3, t], 23], [[s, a0], 20], [[s, a2], 22]], 86]

MFMC cut mask11 OUTWARD capacity := sum(20, 21, 23, 22)

MFMC cut mask11 OUTWARD capacity = 86

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

derived = [[1, 2, 4, 7], [], [[[a0, t], 20], [[a1, t], 21], [[a3, t], 23], [[s, a2], 22]], 86]

MFMC cut mask12 OUTWARD capacity := sum(22, 23, 20, 21)

MFMC cut mask12 OUTWARD capacity = 86

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

derived = [[3, 4, 5, 6], [0], [[[a2, t], 22], [[a3, t], 23], [[s, a0], 20], [[s, a1], 21]], 86]

MFMC cut mask13 OUTWARD capacity := sum(20, 22, 23, 21)

MFMC cut mask13 OUTWARD capacity = 86

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

derived = [[1, 3, 4, 6], [], [[[a0, t], 20], [[a2, t], 22], [[a3, t], 23], [[s, a1], 21]], 86]

MFMC cut mask14 OUTWARD capacity := sum(21, 22, 23, 20)

MFMC cut mask14 OUTWARD capacity = 86

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

derived = [[2, 3, 4, 5], [0], [[[a1, t], 21], [[a2, t], 22], [[a3, t], 23], [[s, a0], 20]], 86]

MFMC cut mask15 OUTWARD capacity := sum(20, 21, 22, 23)

MFMC cut mask15 OUTWARD capacity = 86

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

derived = [[1, 2, 3, 4], [], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[a3, t], 23]], 86]

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

derived = [16, 16, 86, [[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(19)

MFMC vertex a0 ALL incoming = 19

MFMC vertex a0 ALL outgoing := sum(0, 19)

MFMC vertex a0 ALL outgoing = 19

MFMC vertex a1 ALL incoming := sum(20)

MFMC vertex a1 ALL incoming = 20

MFMC vertex a1 ALL outgoing := sum(20)

MFMC vertex a1 ALL outgoing = 20

MFMC vertex a2 ALL incoming := sum(21)

MFMC vertex a2 ALL incoming = 21

MFMC vertex a2 ALL outgoing := sum(21)

MFMC vertex a2 ALL outgoing = 21

MFMC vertex a3 ALL incoming := sum(22)

MFMC vertex a3 ALL incoming = 22

MFMC vertex a3 ALL outgoing := sum(22)

MFMC vertex a3 ALL outgoing = 22

MFMC vertex s ALL incoming := sum(0)

MFMC vertex s ALL incoming = 0

MFMC vertex s ALL outgoing := sum(19, 20, 21, 22)

MFMC vertex s ALL outgoing = 82

MFMC vertex t ALL incoming := sum(19, 20, 21, 22)

MFMC vertex t ALL incoming = 82

MFMC vertex t ALL outgoing := sum()

MFMC vertex t ALL outgoing = 0

MFMC SOURCE net out not gross := sub(82, 0)

MFMC SOURCE net out not gross = 82

MFMC SINK net in not gross := sub(82, 0)

MFMC SINK net in not gross = 82

source roles = [[[a0, s], [a0, t], [a1, t], [a2, t], [a3, t], [s, a0], [s, a1], [s, a2], [s, a3]], [0, 19, 20, 21, 22, 19, 20, 21, 22], [25, 20, 21, 22, 23, 20, 21, 22, 23], s, t]

derived = [[[[a0, s], 0, 25, Yes], [[a0, t], 19, 20, Yes], [[a1, t], 20, 21, Yes], [[a2, t], 21, 22, Yes], [[a3, t], 22, 23, Yes], [[s, a0], 19, 20, Yes], [[s, a1], 20, 21, Yes], [[s, a2], 21, 22, Yes], [[s, a3], 22, 23, Yes]], [[a0, 19, 19, Yes], [a1, 20, 20, Yes], [a2, 21, 21, Yes], [a3, 22, 22, Yes]], Yes, Yes, 82, 82, Yes, Yes]

MFMC cut mask0 OUTWARD flow := sum(19, 20, 21, 22)

MFMC cut mask0 OUTWARD flow = 82

MFMC cut mask0 INWARD flow := sum(0)

MFMC cut mask0 INWARD flow = 0

MFMC cut mask0 NET flow := sub(82, 0)

MFMC cut mask0 NET flow = 82

source roles = [[s], [5, 6, 7, 8], [0], 82]

derived = [[[s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask1 OUTWARD flow := sum(19, 20, 21, 22)

MFMC cut mask1 OUTWARD flow = 82

MFMC cut mask1 INWARD flow := sum()

MFMC cut mask1 INWARD flow = 0

MFMC cut mask1 NET flow := sub(82, 0)

MFMC cut mask1 NET flow = 82

source roles = [[a0, s], [1, 6, 7, 8], [], 82]

derived = [[[a0, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask2 OUTWARD flow := sum(20, 19, 21, 22)

MFMC cut mask2 OUTWARD flow = 82

MFMC cut mask2 INWARD flow := sum(0)

MFMC cut mask2 INWARD flow = 0

MFMC cut mask2 NET flow := sub(82, 0)

MFMC cut mask2 NET flow = 82

source roles = [[a1, s], [2, 5, 7, 8], [0], 82]

derived = [[[a1, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask3 OUTWARD flow := sum(19, 20, 21, 22)

MFMC cut mask3 OUTWARD flow = 82

MFMC cut mask3 INWARD flow := sum()

MFMC cut mask3 INWARD flow = 0

MFMC cut mask3 NET flow := sub(82, 0)

MFMC cut mask3 NET flow = 82

source roles = [[a0, a1, s], [1, 2, 7, 8], [], 82]

derived = [[[a0, a1, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask4 OUTWARD flow := sum(21, 19, 20, 22)

MFMC cut mask4 OUTWARD flow = 82

MFMC cut mask4 INWARD flow := sum(0)

MFMC cut mask4 INWARD flow = 0

MFMC cut mask4 NET flow := sub(82, 0)

MFMC cut mask4 NET flow = 82

source roles = [[a2, s], [3, 5, 6, 8], [0], 82]

derived = [[[a2, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask5 OUTWARD flow := sum(19, 21, 20, 22)

MFMC cut mask5 OUTWARD flow = 82

MFMC cut mask5 INWARD flow := sum()

MFMC cut mask5 INWARD flow = 0

MFMC cut mask5 NET flow := sub(82, 0)

MFMC cut mask5 NET flow = 82

source roles = [[a0, a2, s], [1, 3, 6, 8], [], 82]

derived = [[[a0, a2, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask6 OUTWARD flow := sum(20, 21, 19, 22)

MFMC cut mask6 OUTWARD flow = 82

MFMC cut mask6 INWARD flow := sum(0)

MFMC cut mask6 INWARD flow = 0

MFMC cut mask6 NET flow := sub(82, 0)

MFMC cut mask6 NET flow = 82

source roles = [[a1, a2, s], [2, 3, 5, 8], [0], 82]

derived = [[[a1, a2, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask7 OUTWARD flow := sum(19, 20, 21, 22)

MFMC cut mask7 OUTWARD flow = 82

MFMC cut mask7 INWARD flow := sum()

MFMC cut mask7 INWARD flow = 0

MFMC cut mask7 NET flow := sub(82, 0)

MFMC cut mask7 NET flow = 82

source roles = [[a0, a1, a2, s], [1, 2, 3, 8], [], 82]

derived = [[[a0, a1, a2, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask8 OUTWARD flow := sum(22, 19, 20, 21)

MFMC cut mask8 OUTWARD flow = 82

MFMC cut mask8 INWARD flow := sum(0)

MFMC cut mask8 INWARD flow = 0

MFMC cut mask8 NET flow := sub(82, 0)

MFMC cut mask8 NET flow = 82

source roles = [[a3, s], [4, 5, 6, 7], [0], 82]

derived = [[[a3, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask9 OUTWARD flow := sum(19, 22, 20, 21)

MFMC cut mask9 OUTWARD flow = 82

MFMC cut mask9 INWARD flow := sum()

MFMC cut mask9 INWARD flow = 0

MFMC cut mask9 NET flow := sub(82, 0)

MFMC cut mask9 NET flow = 82

source roles = [[a0, a3, s], [1, 4, 6, 7], [], 82]

derived = [[[a0, a3, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask10 OUTWARD flow := sum(20, 22, 19, 21)

MFMC cut mask10 OUTWARD flow = 82

MFMC cut mask10 INWARD flow := sum(0)

MFMC cut mask10 INWARD flow = 0

MFMC cut mask10 NET flow := sub(82, 0)

MFMC cut mask10 NET flow = 82

source roles = [[a1, a3, s], [2, 4, 5, 7], [0], 82]

derived = [[[a1, a3, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask11 OUTWARD flow := sum(19, 20, 22, 21)

MFMC cut mask11 OUTWARD flow = 82

MFMC cut mask11 INWARD flow := sum()

MFMC cut mask11 INWARD flow = 0

MFMC cut mask11 NET flow := sub(82, 0)

MFMC cut mask11 NET flow = 82

source roles = [[a0, a1, a3, s], [1, 2, 4, 7], [], 82]

derived = [[[a0, a1, a3, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask12 OUTWARD flow := sum(21, 22, 19, 20)

MFMC cut mask12 OUTWARD flow = 82

MFMC cut mask12 INWARD flow := sum(0)

MFMC cut mask12 INWARD flow = 0

MFMC cut mask12 NET flow := sub(82, 0)

MFMC cut mask12 NET flow = 82

source roles = [[a2, a3, s], [3, 4, 5, 6], [0], 82]

derived = [[[a2, a3, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask13 OUTWARD flow := sum(19, 21, 22, 20)

MFMC cut mask13 OUTWARD flow = 82

MFMC cut mask13 INWARD flow := sum()

MFMC cut mask13 INWARD flow = 0

MFMC cut mask13 NET flow := sub(82, 0)

MFMC cut mask13 NET flow = 82

source roles = [[a0, a2, a3, s], [1, 3, 4, 6], [], 82]

derived = [[[a0, a2, a3, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask14 OUTWARD flow := sum(20, 21, 22, 19)

MFMC cut mask14 OUTWARD flow = 82

MFMC cut mask14 INWARD flow := sum(0)

MFMC cut mask14 INWARD flow = 0

MFMC cut mask14 NET flow := sub(82, 0)

MFMC cut mask14 NET flow = 82

source roles = [[a1, a2, a3, s], [2, 3, 4, 5], [0], 82]

derived = [[[a1, a2, a3, s], 82, 0, 82, 86, Yes, Yes]]

MFMC cut mask15 OUTWARD flow := sum(19, 20, 21, 22)

MFMC cut mask15 OUTWARD flow = 82

MFMC cut mask15 INWARD flow := sum()

MFMC cut mask15 INWARD flow = 0

MFMC cut mask15 NET flow := sub(82, 0)

MFMC cut mask15 NET flow = 82

source roles = [[a0, a1, a2, a3, s], [1, 2, 3, 4], [], 82]

derived = [[[a0, a1, a2, a3, s], 82, 0, 82, 86, 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

(82, 82, [[[s], 82, 0, 82, 86, Yes, Yes], [[a0, s], 82, 0, 82, 86, Yes, Yes], [[a1, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, s], 82, 0, 82, 86, Yes, Yes], [[a2, s], 82, 0, 82, 86, Yes, Yes], [[a0, a2, s], 82, 0, 82, 86, Yes, Yes], [[a1, a2, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a2, s], 82, 0, 82, 86, Yes, Yes], [[a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a3, s], 82, 0, 82, 86, Yes, Yes], [[a1, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a3, s], 82, 0, 82, 86, Yes, Yes], [[a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a1, a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a2, a3, s], 82, 0, 82, 86, 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.