# Question 25922: Every Source-Side Vertex Subset Defines One Directed Cut Audit [Return to catalogue page 130](https://ocme.evemisslab.com/en/questions/page/130/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations/#q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0022) 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. R0050-25922 · q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0022 ## Original question and givens Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[23, 23, 24, 24, 25, 25, 28]. 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, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[23, 23, 24, 24, 25, 25, 28]. 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](https://ocme.evemisslab.com/en/mko/mko-max-flow-min-cut-theorem/) ## Step-by-step derivation S001 · impliesBefore source roles = [mfmc-complete-cut-enumeration] After derived = [[vertices, source, sink, cut_rows, cut_count, required_cut_count, 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, a2, t], [[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], [23, 23, 24, 24, 25, 25, 28], null, s, t] After derived = [[a0, a1, a2, s, t], [[a0, s], [a0, t], [a1, t], [a2, t], [s, a0], [s, a1], [s, a2]], [28, 23, 24, 25, 23, 24, 25], null] 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(23, 24, 25) After MFMC cut mask0 OUTWARD capacity = 72 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, a2], [s]] After derived = [[4, 5, 6], [0], [[[s, a0], 23], [[s, a1], 24], [[s, a2], 25]], 72] 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(23, 24, 25) After MFMC cut mask1 OUTWARD capacity = 72 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, a2], [a0, s]] After derived = [[1, 5, 6], [], [[[a0, t], 23], [[s, a1], 24], [[s, a2], 25]], 72] 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(24, 23, 25) After MFMC cut mask2 OUTWARD capacity = 72 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, a2], [a1, s]] After derived = [[2, 4, 6], [0], [[[a1, t], 24], [[s, a0], 23], [[s, a2], 25]], 72] 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(23, 24, 25) After MFMC cut mask3 OUTWARD capacity = 72 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, a2], [a0, a1, s]] After derived = [[1, 2, 6], [], [[[a0, t], 23], [[a1, t], 24], [[s, a2], 25]], 72] 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 · calculationBefore MFMC cut mask4 OUTWARD capacity := sum(25, 23, 24) After MFMC cut mask4 OUTWARD capacity = 72 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, S010 S012 · impliesBefore source roles = [4, [a0, a1, a2], [a2, s]] After derived = [[3, 4, 5], [0], [[[a2, t], 25], [[s, a0], 23], [[s, a1], 24]], 72] 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, S011 S013 · calculationBefore MFMC cut mask5 OUTWARD capacity := sum(23, 25, 24) After MFMC cut mask5 OUTWARD capacity = 72 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 · impliesBefore source roles = [5, [a0, a1, a2], [a0, a2, s]] After derived = [[1, 3, 5], [], [[[a0, t], 23], [[a2, t], 25], [[s, a1], 24]], 72] 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, S013 S015 · calculationBefore MFMC cut mask6 OUTWARD capacity := sum(24, 25, 23) After MFMC cut mask6 OUTWARD capacity = 72 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 · impliesBefore source roles = [6, [a0, a1, a2], [a1, a2, s]] After derived = [[2, 3, 4], [0], [[[a1, t], 24], [[a2, t], 25], [[s, a0], 23]], 72] 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, S015 S017 · calculationBefore MFMC cut mask7 OUTWARD capacity := sum(23, 24, 25) After MFMC cut mask7 OUTWARD capacity = 72 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 · impliesBefore source roles = [7, [a0, a1, a2], [a0, a1, a2, s]] After derived = [[1, 2, 3], [], [[[a0, t], 23], [[a1, t], 24], [[a2, t], 25]], 72] 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, S017 S019 · impliesBefore source roles = [[a0, a1, a2, s, t], [a0, a1, a2]] After derived = [8, 8, 72, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, 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, S018 ## Reverse checks - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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, s, t], s, t, [[[s], [[[s, a0], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a0, s], [[[a0, t], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a1, s], [[[a1, t], 24], [[s, a0], 23], [[s, a2], 25]], 72], [[a0, a1, s], [[[a0, t], 23], [[a1, t], 24], [[s, a2], 25]], 72], [[a2, s], [[[a2, t], 25], [[s, a0], 23], [[s, a1], 24]], 72], [[a0, a2, s], [[[a0, t], 23], [[a2, t], 25], [[s, a1], 24]], 72], [[a1, a2, s], [[[a1, t], 24], [[a2, t], 25], [[s, a0], 23]], 72], [[a0, a1, a2, s], [[[a0, t], 23], [[a1, t], 24], [[a2, t], 25]], 72]], 8, 8, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 26785fa150b65b8d869cbb7e8f84d27acb396dcbc95ae077df18e54beaf1e150 - candidate record SHA-256: 67bfe2287abd40aefdbbc770506844193ab0b661982d8c57a98b3efd378623d4 - semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82 - question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0022.json)