Question 25911: Every Source-Side Vertex Subset Defines One Directed Cut Audit

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=[12, 12, 13, 13, 14, 14, 15, 15]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, 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=[12, 12, 13, 13, 14, 14, 15, 15]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, 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-complete-cut-enumeration]

derived = [[vertices, source, sink, cut_rows, cut_count, required_cut_count, 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]], [12, 12, 13, 13, 14, 14, 15, 15], null, 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]], [12, 13, 14, 15, 12, 13, 14, 15], null]

MFMC cut mask0 OUTWARD capacity := sum(12, 13, 14, 15)

MFMC cut mask0 OUTWARD capacity = 54

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

derived = [[4, 5, 6, 7], [], [[[s, a0], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54]

MFMC cut mask1 OUTWARD capacity := sum(12, 13, 14, 15)

MFMC cut mask1 OUTWARD capacity = 54

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

derived = [[0, 5, 6, 7], [], [[[a0, t], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54]

MFMC cut mask2 OUTWARD capacity := sum(13, 12, 14, 15)

MFMC cut mask2 OUTWARD capacity = 54

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

derived = [[1, 4, 6, 7], [], [[[a1, t], 13], [[s, a0], 12], [[s, a2], 14], [[s, a3], 15]], 54]

MFMC cut mask3 OUTWARD capacity := sum(12, 13, 14, 15)

MFMC cut mask3 OUTWARD capacity = 54

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

derived = [[0, 1, 6, 7], [], [[[a0, t], 12], [[a1, t], 13], [[s, a2], 14], [[s, a3], 15]], 54]

MFMC cut mask4 OUTWARD capacity := sum(14, 12, 13, 15)

MFMC cut mask4 OUTWARD capacity = 54

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

derived = [[2, 4, 5, 7], [], [[[a2, t], 14], [[s, a0], 12], [[s, a1], 13], [[s, a3], 15]], 54]

MFMC cut mask5 OUTWARD capacity := sum(12, 14, 13, 15)

MFMC cut mask5 OUTWARD capacity = 54

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

derived = [[0, 2, 5, 7], [], [[[a0, t], 12], [[a2, t], 14], [[s, a1], 13], [[s, a3], 15]], 54]

MFMC cut mask6 OUTWARD capacity := sum(13, 14, 12, 15)

MFMC cut mask6 OUTWARD capacity = 54

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

derived = [[1, 2, 4, 7], [], [[[a1, t], 13], [[a2, t], 14], [[s, a0], 12], [[s, a3], 15]], 54]

MFMC cut mask7 OUTWARD capacity := sum(12, 13, 14, 15)

MFMC cut mask7 OUTWARD capacity = 54

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

derived = [[0, 1, 2, 7], [], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[s, a3], 15]], 54]

MFMC cut mask8 OUTWARD capacity := sum(15, 12, 13, 14)

MFMC cut mask8 OUTWARD capacity = 54

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

derived = [[3, 4, 5, 6], [], [[[a3, t], 15], [[s, a0], 12], [[s, a1], 13], [[s, a2], 14]], 54]

MFMC cut mask9 OUTWARD capacity := sum(12, 15, 13, 14)

MFMC cut mask9 OUTWARD capacity = 54

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

derived = [[0, 3, 5, 6], [], [[[a0, t], 12], [[a3, t], 15], [[s, a1], 13], [[s, a2], 14]], 54]

MFMC cut mask10 OUTWARD capacity := sum(13, 15, 12, 14)

MFMC cut mask10 OUTWARD capacity = 54

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

derived = [[1, 3, 4, 6], [], [[[a1, t], 13], [[a3, t], 15], [[s, a0], 12], [[s, a2], 14]], 54]

MFMC cut mask11 OUTWARD capacity := sum(12, 13, 15, 14)

MFMC cut mask11 OUTWARD capacity = 54

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

derived = [[0, 1, 3, 6], [], [[[a0, t], 12], [[a1, t], 13], [[a3, t], 15], [[s, a2], 14]], 54]

MFMC cut mask12 OUTWARD capacity := sum(14, 15, 12, 13)

MFMC cut mask12 OUTWARD capacity = 54

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

derived = [[2, 3, 4, 5], [], [[[a2, t], 14], [[a3, t], 15], [[s, a0], 12], [[s, a1], 13]], 54]

MFMC cut mask13 OUTWARD capacity := sum(12, 14, 15, 13)

MFMC cut mask13 OUTWARD capacity = 54

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

derived = [[0, 2, 3, 5], [], [[[a0, t], 12], [[a2, t], 14], [[a3, t], 15], [[s, a1], 13]], 54]

MFMC cut mask14 OUTWARD capacity := sum(13, 14, 15, 12)

MFMC cut mask14 OUTWARD capacity = 54

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

derived = [[1, 2, 3, 4], [], [[[a1, t], 13], [[a2, t], 14], [[a3, t], 15], [[s, a0], 12]], 54]

MFMC cut mask15 OUTWARD capacity := sum(12, 13, 14, 15)

MFMC cut mask15 OUTWARD capacity = 54

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

derived = [[0, 1, 2, 3], [], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[a3, t], 15]], 54]

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

derived = [16, 16, 54, [[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]]]

Reverse checks

• 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

([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, s], [[[a0, t], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a1, s], [[[a1, t], 13], [[s, a0], 12], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, a1, s], [[[a0, t], 12], [[a1, t], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a2, s], [[[a2, t], 14], [[s, a0], 12], [[s, a1], 13], [[s, a3], 15]], 54], [[a0, a2, s], [[[a0, t], 12], [[a2, t], 14], [[s, a1], 13], [[s, a3], 15]], 54], [[a1, a2, s], [[[a1, t], 13], [[a2, t], 14], [[s, a0], 12], [[s, a3], 15]], 54], [[a0, a1, a2, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[s, a3], 15]], 54], [[a3, s], [[[a3, t], 15], [[s, a0], 12], [[s, a1], 13], [[s, a2], 14]], 54], [[a0, a3, s], [[[a0, t], 12], [[a3, t], 15], [[s, a1], 13], [[s, a2], 14]], 54], [[a1, a3, s], [[[a1, t], 13], [[a3, t], 15], [[s, a0], 12], [[s, a2], 14]], 54], [[a0, a1, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a3, t], 15], [[s, a2], 14]], 54], [[a2, a3, s], [[[a2, t], 14], [[a3, t], 15], [[s, a0], 12], [[s, a1], 13]], 54], [[a0, a2, a3, s], [[[a0, t], 12], [[a2, t], 14], [[a3, t], 15], [[s, a1], 13]], 54], [[a1, a2, a3, s], [[[a1, t], 13], [[a2, t], 14], [[a3, t], 15], [[s, a0], 12]], 54], [[a0, a1, a2, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[a3, t], 15]], 54]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

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.