# Complete Directed Cuts and Equal-Value Flow Optimality

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations · 2026-10-15 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Every Source-Side Vertex Subset Defines One Directed Cut Audit

Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[2, 2, 3, 3, 7]. 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).

View answer and explanation([a0, a1, s, t], s, t, [[[s], [[[s, a0], 2], [[s, a1], 3]], 5], [[a0, s], [[[a0, t], 2], [[s, a1], 3]], 5], [[a1, s], [[[a1, t], 3], [[s, a0], 2]], 5], [[a0, a1, s], [[[a0, t], 2], [[a1, t], 3]], 5]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 2], [[s, a1], 3]], 5], [[a0, s], [[[a0, t], 2], [[s, a1], 3]], 5], [[a1, s], [[[a1, t], 3], [[s, a0], 2]], 5], [[a0, a1, s], [[[a0, t], 2], [[a1, t], 3]], 5]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0001/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 2. Every Source-Side Vertex Subset Defines One Directed Cut Audit

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[3, 3, 4, 4, 5, 5]. 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).

View answer and explanation([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a0, s], [[[a0, t], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a1, s], [[[a1, t], 4], [[s, a0], 3], [[s, a2], 5]], 12], [[a0, a1, s], [[[a0, t], 3], [[a1, t], 4], [[s, a2], 5]], 12], [[a2, s], [[[a2, t], 5], [[s, a0], 3], [[s, a1], 4]], 12], [[a0, a2, s], [[[a0, t], 3], [[a2, t], 5], [[s, a1], 4]], 12], [[a1, a2, s], [[[a1, t], 4], [[a2, t], 5], [[s, a0], 3]], 12], [[a0, a1, a2, s], [[[a0, t], 3], [[a1, t], 4], [[a2, t], 5]], 12]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a0, s], [[[a0, t], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a1, s], [[[a1, t], 4], [[s, a0], 3], [[s, a2], 5]], 12], [[a0, a1, s], [[[a0, t], 3], [[a1, t], 4], [[s, a2], 5]], 12], [[a2, s], [[[a2, t], 5], [[s, a0], 3], [[s, a1], 4]], 12], [[a0, a2, s], [[[a0, t], 3], [[a2, t], 5], [[s, a1], 4]], 12], [[a1, a2, s], [[[a1, t], 4], [[a2, t], 5], [[s, a0], 3]], 12], [[a0, a1, a2, s], [[[a0, t], 3], [[a1, t], 4], [[a2, t], 5]], 12]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0002/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 3. Every Source-Side Vertex Subset Defines One Directed Cut Audit

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]. 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).

View answer and explanation([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, s], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a1, s], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, a1, s], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a2, s], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22], [[a0, a2, s], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22], [[a1, a2, s], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22], [[a0, a1, a2, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22], [[a3, s], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22], [[a0, a3, s], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22], [[a1, a3, s], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22], [[a0, a1, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22], [[a2, a3, s], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22], [[a0, a2, a3, s], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22], [[a1, a2, a3, s], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22], [[a0, a1, a2, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, s], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a1, s], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, a1, s], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a2, s], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22], [[a0, a2, s], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22], [[a1, a2, s], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22], [[a0, a1, a2, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22], [[a3, s], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22], [[a0, a3, s], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22], [[a1, a3, s], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22], [[a0, a1, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22], [[a2, a3, s], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22], [[a0, a2, a3, s], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22], [[a1, a2, a3, s], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22], [[a0, a1, a2, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0003/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 4. Every Source-Side Vertex Subset Defines One Directed Cut Audit

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10]. 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).

View answer and explanation([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, s], [[[a0, t], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, s], [[[a1, t], 6], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, s], [[[a0, t], 5], [[a1, t], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a2, s], [[[a2, t], 7], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a2, s], [[[a0, t], 5], [[a2, t], 7], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, a2, s], [[[a1, t], 6], [[a2, t], 7], [[s, a0], 5], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, a2, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a3, s], [[[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a3, s], [[[a0, t], 5], [[a3, t], 8], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a1, a3, s], [[[a1, t], 6], [[a3, t], 8], [[s, a0], 5], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a1, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[s, a2], 7], [[s, a4], 9]], 35], [[a2, a3, s], [[[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a4], 9]], 35], [[a0, a2, a3, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[s, a1], 6], [[s, a4], 9]], 35], [[a1, a2, a3, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a4], 9]], 35], [[a0, a1, a2, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a4], 9]], 35], [[a4, s], [[[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a4, s], [[[a0, t], 5], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a1, a4, s], [[[a1, t], 6], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a1, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a4, t], 9], [[s, a2], 7], [[s, a3], 8]], 35], [[a2, a4, s], [[[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8]], 35], [[a0, a2, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a4, t], 9], [[s, a1], 6], [[s, a3], 8]], 35], [[a1, a2, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a3], 8]], 35], [[a0, a1, a2, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a3], 8]], 35], [[a3, a4, s], [[[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7]], 35], [[a0, a3, a4, s], [[[a0, t], 5], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7]], 35], [[a1, a3, a4, s], [[[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7]], 35], [[a0, a1, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a2], 7]], 35], [[a2, a3, a4, s], [[[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6]], 35], [[a0, a2, a3, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6]], 35], [[a1, a2, a3, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5]], 35], [[a0, a1, a2, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9]], 35]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, s], [[[a0, t], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, s], [[[a1, t], 6], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, s], [[[a0, t], 5], [[a1, t], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a2, s], [[[a2, t], 7], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a2, s], [[[a0, t], 5], [[a2, t], 7], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, a2, s], [[[a1, t], 6], [[a2, t], 7], [[s, a0], 5], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, a2, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a3, s], [[[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a3, s], [[[a0, t], 5], [[a3, t], 8], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a1, a3, s], [[[a1, t], 6], [[a3, t], 8], [[s, a0], 5], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a1, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[s, a2], 7], [[s, a4], 9]], 35], [[a2, a3, s], [[[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a4], 9]], 35], [[a0, a2, a3, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[s, a1], 6], [[s, a4], 9]], 35], [[a1, a2, a3, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a4], 9]], 35], [[a0, a1, a2, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a4], 9]], 35], [[a4, s], [[[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a4, s], [[[a0, t], 5], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a1, a4, s], [[[a1, t], 6], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a1, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a4, t], 9], [[s, a2], 7], [[s, a3], 8]], 35], [[a2, a4, s], [[[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8]], 35], [[a0, a2, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a4, t], 9], [[s, a1], 6], [[s, a3], 8]], 35], [[a1, a2, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a3], 8]], 35], [[a0, a1, a2, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a3], 8]], 35], [[a3, a4, s], [[[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7]], 35], [[a0, a3, a4, s], [[[a0, t], 5], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7]], 35], [[a1, a3, a4, s], [[[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7]], 35], [[a0, a1, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a2], 7]], 35], [[a2, a3, a4, s], [[[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6]], 35], [[a0, a2, a3, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6]], 35], [[a1, a2, a3, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5]], 35], [[a0, a1, a2, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9]], 35]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0004/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 5. Every Source-Side Vertex Subset Defines One Directed Cut Audit

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]. 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).

View answer and explanation([a0, a1, s, t], s, t, [[[s], [[[s, a0], 6], [[s, a1], 7]], 13], [[a0, s], [[[a0, t], 6], [[s, a1], 7]], 13], [[a1, s], [[[a1, t], 7], [[s, a0], 6]], 13], [[a0, a1, s], [[[a0, t], 6], [[a1, t], 7]], 13]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 6], [[s, a1], 7]], 13], [[a0, s], [[[a0, t], 6], [[s, a1], 7]], 13], [[a1, s], [[[a1, t], 7], [[s, a0], 6]], 13], [[a0, a1, s], [[[a0, t], 6], [[a1, t], 7]], 13]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0005/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 6. Every Source-Side Vertex Subset Defines One Directed Cut Audit

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[7, 7, 8, 8, 9, 9]. 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).

View answer and explanation([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a0, s], [[[a0, t], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a1, s], [[[a1, t], 8], [[s, a0], 7], [[s, a2], 9]], 24], [[a0, a1, s], [[[a0, t], 7], [[a1, t], 8], [[s, a2], 9]], 24], [[a2, s], [[[a2, t], 9], [[s, a0], 7], [[s, a1], 8]], 24], [[a0, a2, s], [[[a0, t], 7], [[a2, t], 9], [[s, a1], 8]], 24], [[a1, a2, s], [[[a1, t], 8], [[a2, t], 9], [[s, a0], 7]], 24], [[a0, a1, a2, s], [[[a0, t], 7], [[a1, t], 8], [[a2, t], 9]], 24]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a0, s], [[[a0, t], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a1, s], [[[a1, t], 8], [[s, a0], 7], [[s, a2], 9]], 24], [[a0, a1, s], [[[a0, t], 7], [[a1, t], 8], [[s, a2], 9]], 24], [[a2, s], [[[a2, t], 9], [[s, a0], 7], [[s, a1], 8]], 24], [[a0, a2, s], [[[a0, t], 7], [[a2, t], 9], [[s, a1], 8]], 24], [[a1, a2, s], [[[a1, t], 8], [[a2, t], 9], [[s, a0], 7]], 24], [[a0, a1, a2, s], [[[a0, t], 7], [[a1, t], 8], [[a2, t], 9]], 24]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0006/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 7. Every Source-Side Vertex Subset Defines One Directed Cut Audit

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=[8, 8, 9, 9, 10, 10, 11, 11, 13]. 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).

View answer and explanation([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, s], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a1, s], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, a1, s], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a2, s], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38], [[a0, a2, s], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38], [[a1, a2, s], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38], [[a0, a1, a2, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38], [[a3, s], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38], [[a0, a3, s], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38], [[a1, a3, s], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38], [[a0, a1, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38], [[a2, a3, s], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38], [[a0, a2, a3, s], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38], [[a1, a2, a3, s], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38], [[a0, a1, a2, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, s], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a1, s], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, a1, s], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a2, s], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38], [[a0, a2, s], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38], [[a1, a2, s], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38], [[a0, a1, a2, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38], [[a3, s], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38], [[a0, a3, s], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38], [[a1, a3, s], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38], [[a0, a1, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38], [[a2, a3, s], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38], [[a0, a2, a3, s], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38], [[a1, a2, a3, s], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38], [[a0, a1, a2, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0007/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 8. Every Source-Side Vertex Subset Defines One Directed Cut Audit

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13]. 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).

View answer and explanation([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0008/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 9. Every Source-Side Vertex Subset Defines One Directed Cut Audit

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=[10, 10, 11, 11]. 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).

View answer and explanation([a0, a1, s, t], s, t, [[[s], [[[s, a0], 10], [[s, a1], 11]], 21], [[a0, s], [[[a0, t], 10], [[s, a1], 11]], 21], [[a1, s], [[[a1, t], 11], [[s, a0], 10]], 21], [[a0, a1, s], [[[a0, t], 10], [[a1, t], 11]], 21]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 10], [[s, a1], 11]], 21], [[a0, s], [[[a0, t], 10], [[s, a1], 11]], 21], [[a1, s], [[[a1, t], 11], [[s, a0], 10]], 21], [[a0, a1, s], [[[a0, t], 10], [[a1, t], 11]], 21]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0009/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 10. Every Source-Side Vertex Subset Defines One Directed Cut Audit

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=[11, 11, 12, 12, 13, 13, 16]. 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).

View answer and explanation([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a0, s], [[[a0, t], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a1, s], [[[a1, t], 12], [[s, a0], 11], [[s, a2], 13]], 36], [[a0, a1, s], [[[a0, t], 11], [[a1, t], 12], [[s, a2], 13]], 36], [[a2, s], [[[a2, t], 13], [[s, a0], 11], [[s, a1], 12]], 36], [[a0, a2, s], [[[a0, t], 11], [[a2, t], 13], [[s, a1], 12]], 36], [[a1, a2, s], [[[a1, t], 12], [[a2, t], 13], [[s, a0], 11]], 36], [[a0, a1, a2, s], [[[a0, t], 11], [[a1, t], 12], [[a2, t], 13]], 36]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a0, s], [[[a0, t], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a1, s], [[[a1, t], 12], [[s, a0], 11], [[s, a2], 13]], 36], [[a0, a1, s], [[[a0, t], 11], [[a1, t], 12], [[s, a2], 13]], 36], [[a2, s], [[[a2, t], 13], [[s, a0], 11], [[s, a1], 12]], 36], [[a0, a2, s], [[[a0, t], 11], [[a2, t], 13], [[s, a1], 12]], 36], [[a1, a2, s], [[[a1, t], 12], [[a2, t], 13], [[s, a0], 11]], 36], [[a0, a1, a2, s], [[[a0, t], 11], [[a1, t], 12], [[a2, t], 13]], 36]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0010/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 11. Every Source-Side Vertex Subset Defines One Directed Cut Audit

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).

View answer and explanation([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)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0011/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 12. Every Source-Side Vertex Subset Defines One Directed Cut Audit

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17]. 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).

View answer and explanation([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, s], [[[a0, t], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, s], [[[a1, t], 14], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, s], [[[a0, t], 13], [[a1, t], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a2, s], [[[a2, t], 15], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a2, s], [[[a0, t], 13], [[a2, t], 15], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, a2, s], [[[a1, t], 14], [[a2, t], 15], [[s, a0], 13], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, a2, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a3, s], [[[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a3, s], [[[a0, t], 13], [[a3, t], 16], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a1, a3, s], [[[a1, t], 14], [[a3, t], 16], [[s, a0], 13], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a1, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[s, a2], 15], [[s, a4], 17]], 75], [[a2, a3, s], [[[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a4], 17]], 75], [[a0, a2, a3, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[s, a1], 14], [[s, a4], 17]], 75], [[a1, a2, a3, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a4], 17]], 75], [[a0, a1, a2, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a4], 17]], 75], [[a4, s], [[[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a4, s], [[[a0, t], 13], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a1, a4, s], [[[a1, t], 14], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a1, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a4, t], 17], [[s, a2], 15], [[s, a3], 16]], 75], [[a2, a4, s], [[[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16]], 75], [[a0, a2, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a4, t], 17], [[s, a1], 14], [[s, a3], 16]], 75], [[a1, a2, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a3], 16]], 75], [[a0, a1, a2, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a3], 16]], 75], [[a3, a4, s], [[[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15]], 75], [[a0, a3, a4, s], [[[a0, t], 13], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15]], 75], [[a1, a3, a4, s], [[[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15]], 75], [[a0, a1, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a2], 15]], 75], [[a2, a3, a4, s], [[[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14]], 75], [[a0, a2, a3, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14]], 75], [[a1, a2, a3, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13]], 75], [[a0, a1, a2, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17]], 75]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, s], [[[a0, t], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, s], [[[a1, t], 14], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, s], [[[a0, t], 13], [[a1, t], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a2, s], [[[a2, t], 15], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a2, s], [[[a0, t], 13], [[a2, t], 15], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, a2, s], [[[a1, t], 14], [[a2, t], 15], [[s, a0], 13], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, a2, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a3, s], [[[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a3, s], [[[a0, t], 13], [[a3, t], 16], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a1, a3, s], [[[a1, t], 14], [[a3, t], 16], [[s, a0], 13], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a1, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[s, a2], 15], [[s, a4], 17]], 75], [[a2, a3, s], [[[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a4], 17]], 75], [[a0, a2, a3, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[s, a1], 14], [[s, a4], 17]], 75], [[a1, a2, a3, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a4], 17]], 75], [[a0, a1, a2, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a4], 17]], 75], [[a4, s], [[[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a4, s], [[[a0, t], 13], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a1, a4, s], [[[a1, t], 14], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a1, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a4, t], 17], [[s, a2], 15], [[s, a3], 16]], 75], [[a2, a4, s], [[[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16]], 75], [[a0, a2, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a4, t], 17], [[s, a1], 14], [[s, a3], 16]], 75], [[a1, a2, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a3], 16]], 75], [[a0, a1, a2, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a3], 16]], 75], [[a3, a4, s], [[[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15]], 75], [[a0, a3, a4, s], [[[a0, t], 13], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15]], 75], [[a1, a3, a4, s], [[[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15]], 75], [[a0, a1, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a2], 15]], 75], [[a2, a3, a4, s], [[[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14]], 75], [[a0, a2, a3, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14]], 75], [[a1, a2, a3, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13]], 75], [[a0, a1, a2, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17]], 75]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0012/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 13. Every Source-Side Vertex Subset Defines One Directed Cut Audit

Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[14, 14, 15, 15, 19]. 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).

View answer and explanation([a0, a1, s, t], s, t, [[[s], [[[s, a0], 14], [[s, a1], 15]], 29], [[a0, s], [[[a0, t], 14], [[s, a1], 15]], 29], [[a1, s], [[[a1, t], 15], [[s, a0], 14]], 29], [[a0, a1, s], [[[a0, t], 14], [[a1, t], 15]], 29]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 14], [[s, a1], 15]], 29], [[a0, s], [[[a0, t], 14], [[s, a1], 15]], 29], [[a1, s], [[[a1, t], 15], [[s, a0], 14]], 29], [[a0, a1, s], [[[a0, t], 14], [[a1, t], 15]], 29]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0013/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 14. Every Source-Side Vertex Subset Defines One Directed Cut Audit

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[15, 15, 16, 16, 17, 17]. 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).

View answer and explanation([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a0, s], [[[a0, t], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a1, s], [[[a1, t], 16], [[s, a0], 15], [[s, a2], 17]], 48], [[a0, a1, s], [[[a0, t], 15], [[a1, t], 16], [[s, a2], 17]], 48], [[a2, s], [[[a2, t], 17], [[s, a0], 15], [[s, a1], 16]], 48], [[a0, a2, s], [[[a0, t], 15], [[a2, t], 17], [[s, a1], 16]], 48], [[a1, a2, s], [[[a1, t], 16], [[a2, t], 17], [[s, a0], 15]], 48], [[a0, a1, a2, s], [[[a0, t], 15], [[a1, t], 16], [[a2, t], 17]], 48]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a0, s], [[[a0, t], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a1, s], [[[a1, t], 16], [[s, a0], 15], [[s, a2], 17]], 48], [[a0, a1, s], [[[a0, t], 15], [[a1, t], 16], [[s, a2], 17]], 48], [[a2, s], [[[a2, t], 17], [[s, a0], 15], [[s, a1], 16]], 48], [[a0, a2, s], [[[a0, t], 15], [[a2, t], 17], [[s, a1], 16]], 48], [[a1, a2, s], [[[a1, t], 16], [[a2, t], 17], [[s, a0], 15]], 48], [[a0, a1, a2, s], [[[a0, t], 15], [[a1, t], 16], [[a2, t], 17]], 48]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0014/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 15. Every Source-Side Vertex Subset Defines One Directed Cut Audit

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=[16, 16, 17, 17, 18, 18, 19, 19]. 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).

View answer and explanation([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, s], [[[a0, t], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a1, s], [[[a1, t], 17], [[s, a0], 16], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, a1, s], [[[a0, t], 16], [[a1, t], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a2, s], [[[a2, t], 18], [[s, a0], 16], [[s, a1], 17], [[s, a3], 19]], 70], [[a0, a2, s], [[[a0, t], 16], [[a2, t], 18], [[s, a1], 17], [[s, a3], 19]], 70], [[a1, a2, s], [[[a1, t], 17], [[a2, t], 18], [[s, a0], 16], [[s, a3], 19]], 70], [[a0, a1, a2, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[s, a3], 19]], 70], [[a3, s], [[[a3, t], 19], [[s, a0], 16], [[s, a1], 17], [[s, a2], 18]], 70], [[a0, a3, s], [[[a0, t], 16], [[a3, t], 19], [[s, a1], 17], [[s, a2], 18]], 70], [[a1, a3, s], [[[a1, t], 17], [[a3, t], 19], [[s, a0], 16], [[s, a2], 18]], 70], [[a0, a1, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a3, t], 19], [[s, a2], 18]], 70], [[a2, a3, s], [[[a2, t], 18], [[a3, t], 19], [[s, a0], 16], [[s, a1], 17]], 70], [[a0, a2, a3, s], [[[a0, t], 16], [[a2, t], 18], [[a3, t], 19], [[s, a1], 17]], 70], [[a1, a2, a3, s], [[[a1, t], 17], [[a2, t], 18], [[a3, t], 19], [[s, a0], 16]], 70], [[a0, a1, a2, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[a3, t], 19]], 70]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, s], [[[a0, t], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a1, s], [[[a1, t], 17], [[s, a0], 16], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, a1, s], [[[a0, t], 16], [[a1, t], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a2, s], [[[a2, t], 18], [[s, a0], 16], [[s, a1], 17], [[s, a3], 19]], 70], [[a0, a2, s], [[[a0, t], 16], [[a2, t], 18], [[s, a1], 17], [[s, a3], 19]], 70], [[a1, a2, s], [[[a1, t], 17], [[a2, t], 18], [[s, a0], 16], [[s, a3], 19]], 70], [[a0, a1, a2, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[s, a3], 19]], 70], [[a3, s], [[[a3, t], 19], [[s, a0], 16], [[s, a1], 17], [[s, a2], 18]], 70], [[a0, a3, s], [[[a0, t], 16], [[a3, t], 19], [[s, a1], 17], [[s, a2], 18]], 70], [[a1, a3, s], [[[a1, t], 17], [[a3, t], 19], [[s, a0], 16], [[s, a2], 18]], 70], [[a0, a1, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a3, t], 19], [[s, a2], 18]], 70], [[a2, a3, s], [[[a2, t], 18], [[a3, t], 19], [[s, a0], 16], [[s, a1], 17]], 70], [[a0, a2, a3, s], [[[a0, t], 16], [[a2, t], 18], [[a3, t], 19], [[s, a1], 17]], 70], [[a1, a2, a3, s], [[[a1, t], 17], [[a2, t], 18], [[a3, t], 19], [[s, a0], 16]], 70], [[a0, a1, a2, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[a3, t], 19]], 70]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0015/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 16. Every Source-Side Vertex Subset Defines One Directed Cut Audit

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22]. 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).

View answer and explanation([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, s], [[[a0, t], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, s], [[[a1, t], 18], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, s], [[[a0, t], 17], [[a1, t], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a2, s], [[[a2, t], 19], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a2, s], [[[a0, t], 17], [[a2, t], 19], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, a2, s], [[[a1, t], 18], [[a2, t], 19], [[s, a0], 17], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, a2, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a3, s], [[[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a3, s], [[[a0, t], 17], [[a3, t], 20], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a1, a3, s], [[[a1, t], 18], [[a3, t], 20], [[s, a0], 17], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a1, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[s, a2], 19], [[s, a4], 21]], 95], [[a2, a3, s], [[[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a4], 21]], 95], [[a0, a2, a3, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[s, a1], 18], [[s, a4], 21]], 95], [[a1, a2, a3, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a4], 21]], 95], [[a0, a1, a2, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a4], 21]], 95], [[a4, s], [[[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a4, s], [[[a0, t], 17], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a1, a4, s], [[[a1, t], 18], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a1, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a4, t], 21], [[s, a2], 19], [[s, a3], 20]], 95], [[a2, a4, s], [[[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20]], 95], [[a0, a2, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a4, t], 21], [[s, a1], 18], [[s, a3], 20]], 95], [[a1, a2, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a3], 20]], 95], [[a0, a1, a2, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a3], 20]], 95], [[a3, a4, s], [[[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19]], 95], [[a0, a3, a4, s], [[[a0, t], 17], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19]], 95], [[a1, a3, a4, s], [[[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19]], 95], [[a0, a1, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a2], 19]], 95], [[a2, a3, a4, s], [[[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18]], 95], [[a0, a2, a3, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18]], 95], [[a1, a2, a3, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17]], 95], [[a0, a1, a2, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21]], 95]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, s], [[[a0, t], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, s], [[[a1, t], 18], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, s], [[[a0, t], 17], [[a1, t], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a2, s], [[[a2, t], 19], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a2, s], [[[a0, t], 17], [[a2, t], 19], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, a2, s], [[[a1, t], 18], [[a2, t], 19], [[s, a0], 17], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, a2, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a3, s], [[[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a3, s], [[[a0, t], 17], [[a3, t], 20], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a1, a3, s], [[[a1, t], 18], [[a3, t], 20], [[s, a0], 17], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a1, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[s, a2], 19], [[s, a4], 21]], 95], [[a2, a3, s], [[[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a4], 21]], 95], [[a0, a2, a3, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[s, a1], 18], [[s, a4], 21]], 95], [[a1, a2, a3, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a4], 21]], 95], [[a0, a1, a2, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a4], 21]], 95], [[a4, s], [[[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a4, s], [[[a0, t], 17], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a1, a4, s], [[[a1, t], 18], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a1, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a4, t], 21], [[s, a2], 19], [[s, a3], 20]], 95], [[a2, a4, s], [[[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20]], 95], [[a0, a2, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a4, t], 21], [[s, a1], 18], [[s, a3], 20]], 95], [[a1, a2, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a3], 20]], 95], [[a0, a1, a2, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a3], 20]], 95], [[a3, a4, s], [[[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19]], 95], [[a0, a3, a4, s], [[[a0, t], 17], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19]], 95], [[a1, a3, a4, s], [[[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19]], 95], [[a0, a1, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a2], 19]], 95], [[a2, a3, a4, s], [[[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18]], 95], [[a0, a2, a3, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18]], 95], [[a1, a2, a3, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17]], 95], [[a0, a1, a2, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21]], 95]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0016/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 17. Every Source-Side Vertex Subset Defines One Directed Cut Audit

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=[18, 18, 19, 19]. 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).

View answer and explanation([a0, a1, s, t], s, t, [[[s], [[[s, a0], 18], [[s, a1], 19]], 37], [[a0, s], [[[a0, t], 18], [[s, a1], 19]], 37], [[a1, s], [[[a1, t], 19], [[s, a0], 18]], 37], [[a0, a1, s], [[[a0, t], 18], [[a1, t], 19]], 37]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 18], [[s, a1], 19]], 37], [[a0, s], [[[a0, t], 18], [[s, a1], 19]], 37], [[a1, s], [[[a1, t], 19], [[s, a0], 18]], 37], [[a0, a1, s], [[[a0, t], 18], [[a1, t], 19]], 37]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0017/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 18. Every Source-Side Vertex Subset Defines One Directed Cut Audit

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[19, 19, 20, 20, 21, 21]. 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).

View answer and explanation([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a0, s], [[[a0, t], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a1, s], [[[a1, t], 20], [[s, a0], 19], [[s, a2], 21]], 60], [[a0, a1, s], [[[a0, t], 19], [[a1, t], 20], [[s, a2], 21]], 60], [[a2, s], [[[a2, t], 21], [[s, a0], 19], [[s, a1], 20]], 60], [[a0, a2, s], [[[a0, t], 19], [[a2, t], 21], [[s, a1], 20]], 60], [[a1, a2, s], [[[a1, t], 20], [[a2, t], 21], [[s, a0], 19]], 60], [[a0, a1, a2, s], [[[a0, t], 19], [[a1, t], 20], [[a2, t], 21]], 60]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a0, s], [[[a0, t], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a1, s], [[[a1, t], 20], [[s, a0], 19], [[s, a2], 21]], 60], [[a0, a1, s], [[[a0, t], 19], [[a1, t], 20], [[s, a2], 21]], 60], [[a2, s], [[[a2, t], 21], [[s, a0], 19], [[s, a1], 20]], 60], [[a0, a2, s], [[[a0, t], 19], [[a2, t], 21], [[s, a1], 20]], 60], [[a1, a2, s], [[[a1, t], 20], [[a2, t], 21], [[s, a0], 19]], 60], [[a0, a1, a2, s], [[[a0, t], 19], [[a1, t], 20], [[a2, t], 21]], 60]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0018/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 19. Every Source-Side Vertex Subset Defines One Directed Cut Audit

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]. 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).

View answer and explanation([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, s], [[[a0, t], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a1, s], [[[a1, t], 21], [[s, a0], 20], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, a1, s], [[[a0, t], 20], [[a1, t], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a2, s], [[[a2, t], 22], [[s, a0], 20], [[s, a1], 21], [[s, a3], 23]], 86], [[a0, a2, s], [[[a0, t], 20], [[a2, t], 22], [[s, a1], 21], [[s, a3], 23]], 86], [[a1, a2, s], [[[a1, t], 21], [[a2, t], 22], [[s, a0], 20], [[s, a3], 23]], 86], [[a0, a1, a2, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[s, a3], 23]], 86], [[a3, s], [[[a3, t], 23], [[s, a0], 20], [[s, a1], 21], [[s, a2], 22]], 86], [[a0, a3, s], [[[a0, t], 20], [[a3, t], 23], [[s, a1], 21], [[s, a2], 22]], 86], [[a1, a3, s], [[[a1, t], 21], [[a3, t], 23], [[s, a0], 20], [[s, a2], 22]], 86], [[a0, a1, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a3, t], 23], [[s, a2], 22]], 86], [[a2, a3, s], [[[a2, t], 22], [[a3, t], 23], [[s, a0], 20], [[s, a1], 21]], 86], [[a0, a2, a3, s], [[[a0, t], 20], [[a2, t], 22], [[a3, t], 23], [[s, a1], 21]], 86], [[a1, a2, a3, s], [[[a1, t], 21], [[a2, t], 22], [[a3, t], 23], [[s, a0], 20]], 86], [[a0, a1, a2, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[a3, t], 23]], 86]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, s], [[[a0, t], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a1, s], [[[a1, t], 21], [[s, a0], 20], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, a1, s], [[[a0, t], 20], [[a1, t], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a2, s], [[[a2, t], 22], [[s, a0], 20], [[s, a1], 21], [[s, a3], 23]], 86], [[a0, a2, s], [[[a0, t], 20], [[a2, t], 22], [[s, a1], 21], [[s, a3], 23]], 86], [[a1, a2, s], [[[a1, t], 21], [[a2, t], 22], [[s, a0], 20], [[s, a3], 23]], 86], [[a0, a1, a2, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[s, a3], 23]], 86], [[a3, s], [[[a3, t], 23], [[s, a0], 20], [[s, a1], 21], [[s, a2], 22]], 86], [[a0, a3, s], [[[a0, t], 20], [[a3, t], 23], [[s, a1], 21], [[s, a2], 22]], 86], [[a1, a3, s], [[[a1, t], 21], [[a3, t], 23], [[s, a0], 20], [[s, a2], 22]], 86], [[a0, a1, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a3, t], 23], [[s, a2], 22]], 86], [[a2, a3, s], [[[a2, t], 22], [[a3, t], 23], [[s, a0], 20], [[s, a1], 21]], 86], [[a0, a2, a3, s], [[[a0, t], 20], [[a2, t], 22], [[a3, t], 23], [[s, a1], 21]], 86], [[a1, a2, a3, s], [[[a1, t], 21], [[a2, t], 22], [[a3, t], 23], [[s, a0], 20]], 86], [[a0, a1, a2, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[a3, t], 23]], 86]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0019/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 20. Every Source-Side Vertex Subset Defines One Directed Cut Audit

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25]. 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).

View answer and explanation([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, s], [[[a0, t], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, s], [[[a1, t], 22], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, s], [[[a0, t], 21], [[a1, t], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a2, s], [[[a2, t], 23], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a2, s], [[[a0, t], 21], [[a2, t], 23], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, a2, s], [[[a1, t], 22], [[a2, t], 23], [[s, a0], 21], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, a2, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a3, s], [[[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a3, s], [[[a0, t], 21], [[a3, t], 24], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a1, a3, s], [[[a1, t], 22], [[a3, t], 24], [[s, a0], 21], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a1, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[s, a2], 23], [[s, a4], 25]], 115], [[a2, a3, s], [[[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a4], 25]], 115], [[a0, a2, a3, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[s, a1], 22], [[s, a4], 25]], 115], [[a1, a2, a3, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a4], 25]], 115], [[a0, a1, a2, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a4], 25]], 115], [[a4, s], [[[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a4, s], [[[a0, t], 21], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a1, a4, s], [[[a1, t], 22], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a1, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a4, t], 25], [[s, a2], 23], [[s, a3], 24]], 115], [[a2, a4, s], [[[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24]], 115], [[a0, a2, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a4, t], 25], [[s, a1], 22], [[s, a3], 24]], 115], [[a1, a2, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a3], 24]], 115], [[a0, a1, a2, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a3], 24]], 115], [[a3, a4, s], [[[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23]], 115], [[a0, a3, a4, s], [[[a0, t], 21], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23]], 115], [[a1, a3, a4, s], [[[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23]], 115], [[a0, a1, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a2], 23]], 115], [[a2, a3, a4, s], [[[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22]], 115], [[a0, a2, a3, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22]], 115], [[a1, a2, a3, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21]], 115], [[a0, a1, a2, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25]], 115]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, s], [[[a0, t], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, s], [[[a1, t], 22], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, s], [[[a0, t], 21], [[a1, t], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a2, s], [[[a2, t], 23], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a2, s], [[[a0, t], 21], [[a2, t], 23], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, a2, s], [[[a1, t], 22], [[a2, t], 23], [[s, a0], 21], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, a2, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a3, s], [[[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a3, s], [[[a0, t], 21], [[a3, t], 24], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a1, a3, s], [[[a1, t], 22], [[a3, t], 24], [[s, a0], 21], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a1, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[s, a2], 23], [[s, a4], 25]], 115], [[a2, a3, s], [[[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a4], 25]], 115], [[a0, a2, a3, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[s, a1], 22], [[s, a4], 25]], 115], [[a1, a2, a3, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a4], 25]], 115], [[a0, a1, a2, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a4], 25]], 115], [[a4, s], [[[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a4, s], [[[a0, t], 21], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a1, a4, s], [[[a1, t], 22], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a1, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a4, t], 25], [[s, a2], 23], [[s, a3], 24]], 115], [[a2, a4, s], [[[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24]], 115], [[a0, a2, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a4, t], 25], [[s, a1], 22], [[s, a3], 24]], 115], [[a1, a2, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a3], 24]], 115], [[a0, a1, a2, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a3], 24]], 115], [[a3, a4, s], [[[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23]], 115], [[a0, a3, a4, s], [[[a0, t], 21], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23]], 115], [[a1, a3, a4, s], [[[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23]], 115], [[a0, a1, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a2], 23]], 115], [[a2, a3, a4, s], [[[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22]], 115], [[a0, a2, a3, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22]], 115], [[a1, a2, a3, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21]], 115], [[a0, a1, a2, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25]], 115]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0020/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 21. Every Source-Side Vertex Subset Defines One Directed Cut Audit

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=[22, 22, 23, 23]. 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).

View answer and explanation([a0, a1, s, t], s, t, [[[s], [[[s, a0], 22], [[s, a1], 23]], 45], [[a0, s], [[[a0, t], 22], [[s, a1], 23]], 45], [[a1, s], [[[a1, t], 23], [[s, a0], 22]], 45], [[a0, a1, s], [[[a0, t], 22], [[a1, t], 23]], 45]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 22], [[s, a1], 23]], 45], [[a0, s], [[[a0, t], 22], [[s, a1], 23]], 45], [[a1, s], [[[a1, t], 23], [[s, a0], 22]], 45], [[a0, a1, s], [[[a0, t], 22], [[a1, t], 23]], 45]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0021/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 22. Every Source-Side Vertex Subset Defines One Directed Cut Audit

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).

View answer and explanation([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)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0022/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 23. Every Source-Side Vertex Subset Defines One Directed Cut Audit

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=[24, 24, 25, 25, 26, 26, 27, 27]. 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).

View answer and explanation([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, s], [[[a0, t], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a1, s], [[[a1, t], 25], [[s, a0], 24], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, a1, s], [[[a0, t], 24], [[a1, t], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a2, s], [[[a2, t], 26], [[s, a0], 24], [[s, a1], 25], [[s, a3], 27]], 102], [[a0, a2, s], [[[a0, t], 24], [[a2, t], 26], [[s, a1], 25], [[s, a3], 27]], 102], [[a1, a2, s], [[[a1, t], 25], [[a2, t], 26], [[s, a0], 24], [[s, a3], 27]], 102], [[a0, a1, a2, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[s, a3], 27]], 102], [[a3, s], [[[a3, t], 27], [[s, a0], 24], [[s, a1], 25], [[s, a2], 26]], 102], [[a0, a3, s], [[[a0, t], 24], [[a3, t], 27], [[s, a1], 25], [[s, a2], 26]], 102], [[a1, a3, s], [[[a1, t], 25], [[a3, t], 27], [[s, a0], 24], [[s, a2], 26]], 102], [[a0, a1, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a3, t], 27], [[s, a2], 26]], 102], [[a2, a3, s], [[[a2, t], 26], [[a3, t], 27], [[s, a0], 24], [[s, a1], 25]], 102], [[a0, a2, a3, s], [[[a0, t], 24], [[a2, t], 26], [[a3, t], 27], [[s, a1], 25]], 102], [[a1, a2, a3, s], [[[a1, t], 25], [[a2, t], 26], [[a3, t], 27], [[s, a0], 24]], 102], [[a0, a1, a2, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[a3, t], 27]], 102]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, s], [[[a0, t], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a1, s], [[[a1, t], 25], [[s, a0], 24], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, a1, s], [[[a0, t], 24], [[a1, t], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a2, s], [[[a2, t], 26], [[s, a0], 24], [[s, a1], 25], [[s, a3], 27]], 102], [[a0, a2, s], [[[a0, t], 24], [[a2, t], 26], [[s, a1], 25], [[s, a3], 27]], 102], [[a1, a2, s], [[[a1, t], 25], [[a2, t], 26], [[s, a0], 24], [[s, a3], 27]], 102], [[a0, a1, a2, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[s, a3], 27]], 102], [[a3, s], [[[a3, t], 27], [[s, a0], 24], [[s, a1], 25], [[s, a2], 26]], 102], [[a0, a3, s], [[[a0, t], 24], [[a3, t], 27], [[s, a1], 25], [[s, a2], 26]], 102], [[a1, a3, s], [[[a1, t], 25], [[a3, t], 27], [[s, a0], 24], [[s, a2], 26]], 102], [[a0, a1, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a3, t], 27], [[s, a2], 26]], 102], [[a2, a3, s], [[[a2, t], 26], [[a3, t], 27], [[s, a0], 24], [[s, a1], 25]], 102], [[a0, a2, a3, s], [[[a0, t], 24], [[a2, t], 26], [[a3, t], 27], [[s, a1], 25]], 102], [[a1, a2, a3, s], [[[a1, t], 25], [[a2, t], 26], [[a3, t], 27], [[s, a0], 24]], 102], [[a0, a1, a2, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[a3, t], 27]], 102]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0023/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 24. Every Source-Side Vertex Subset Defines One Directed Cut Audit

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29]. 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).

View answer and explanation([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, s], [[[a0, t], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, s], [[[a1, t], 26], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, s], [[[a0, t], 25], [[a1, t], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a2, s], [[[a2, t], 27], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a2, s], [[[a0, t], 25], [[a2, t], 27], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, a2, s], [[[a1, t], 26], [[a2, t], 27], [[s, a0], 25], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, a2, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a3, s], [[[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a3, s], [[[a0, t], 25], [[a3, t], 28], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a1, a3, s], [[[a1, t], 26], [[a3, t], 28], [[s, a0], 25], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a1, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[s, a2], 27], [[s, a4], 29]], 135], [[a2, a3, s], [[[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a4], 29]], 135], [[a0, a2, a3, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[s, a1], 26], [[s, a4], 29]], 135], [[a1, a2, a3, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a4], 29]], 135], [[a0, a1, a2, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a4], 29]], 135], [[a4, s], [[[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a4, s], [[[a0, t], 25], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a1, a4, s], [[[a1, t], 26], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a1, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a4, t], 29], [[s, a2], 27], [[s, a3], 28]], 135], [[a2, a4, s], [[[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28]], 135], [[a0, a2, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a4, t], 29], [[s, a1], 26], [[s, a3], 28]], 135], [[a1, a2, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a3], 28]], 135], [[a0, a1, a2, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a3], 28]], 135], [[a3, a4, s], [[[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27]], 135], [[a0, a3, a4, s], [[[a0, t], 25], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27]], 135], [[a1, a3, a4, s], [[[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27]], 135], [[a0, a1, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a2], 27]], 135], [[a2, a3, a4, s], [[[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26]], 135], [[a0, a2, a3, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26]], 135], [[a1, a2, a3, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25]], 135], [[a0, a1, a2, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29]], 135]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, s], [[[a0, t], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, s], [[[a1, t], 26], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, s], [[[a0, t], 25], [[a1, t], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a2, s], [[[a2, t], 27], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a2, s], [[[a0, t], 25], [[a2, t], 27], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, a2, s], [[[a1, t], 26], [[a2, t], 27], [[s, a0], 25], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, a2, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a3, s], [[[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a3, s], [[[a0, t], 25], [[a3, t], 28], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a1, a3, s], [[[a1, t], 26], [[a3, t], 28], [[s, a0], 25], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a1, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[s, a2], 27], [[s, a4], 29]], 135], [[a2, a3, s], [[[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a4], 29]], 135], [[a0, a2, a3, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[s, a1], 26], [[s, a4], 29]], 135], [[a1, a2, a3, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a4], 29]], 135], [[a0, a1, a2, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a4], 29]], 135], [[a4, s], [[[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a4, s], [[[a0, t], 25], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a1, a4, s], [[[a1, t], 26], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a1, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a4, t], 29], [[s, a2], 27], [[s, a3], 28]], 135], [[a2, a4, s], [[[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28]], 135], [[a0, a2, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a4, t], 29], [[s, a1], 26], [[s, a3], 28]], 135], [[a1, a2, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a3], 28]], 135], [[a0, a1, a2, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a3], 28]], 135], [[a3, a4, s], [[[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27]], 135], [[a0, a3, a4, s], [[[a0, t], 25], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27]], 135], [[a1, a3, a4, s], [[[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27]], 135], [[a0, a1, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a2], 27]], 135], [[a2, a3, a4, s], [[[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26]], 135], [[a0, a2, a3, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26]], 135], [[a1, a2, a3, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25]], 135], [[a0, a1, a2, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29]], 135]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0024/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 25. Every Source-Side Vertex Subset Defines One Directed Cut Audit

Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[26, 26, 27, 27, 31]. 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).

View answer and explanation([a0, a1, s, t], s, t, [[[s], [[[s, a0], 26], [[s, a1], 27]], 53], [[a0, s], [[[a0, t], 26], [[s, a1], 27]], 53], [[a1, s], [[[a1, t], 27], [[s, a0], 26]], 53], [[a0, a1, s], [[[a0, t], 26], [[a1, t], 27]], 53]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)

Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 26], [[s, a1], 27]], 53], [[a0, s], [[[a0, t], 26], [[s, a1], 27]], 53], [[a1, s], [[[a1, t], 27], [[s, a0], 26]], 53], [[a0, a1, s], [[[a0, t], 26], [[a1, t], 27]], 53]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0025/)

Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 26. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[2, 2, 3, 3, 7], candidate flows=[1, 1, 2, 2, 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(3, 3, [[[s], 3, 0, 3, 5, Yes, Yes], [[a0, s], 3, 0, 3, 5, Yes, Yes], [[a1, s], 3, 0, 3, 5, Yes, Yes], [[a0, a1, s], 3, 0, 3, 5, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (3, 3, [[[s], 3, 0, 3, 5, Yes, Yes], [[a0, s], 3, 0, 3, 5, Yes, Yes], [[a1, s], 3, 0, 3, 5, Yes, Yes], [[a0, a1, s], 3, 0, 3, 5, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0026/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 27. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[3, 3, 4, 4, 5, 5], candidate flows=[2, 2, 3, 3, 4, 4]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(9, 9, [[[s], 9, 0, 9, 12, Yes, Yes], [[a0, s], 9, 0, 9, 12, Yes, Yes], [[a1, s], 9, 0, 9, 12, Yes, Yes], [[a0, a1, s], 9, 0, 9, 12, Yes, Yes], [[a2, s], 9, 0, 9, 12, Yes, Yes], [[a0, a2, s], 9, 0, 9, 12, Yes, Yes], [[a1, a2, s], 9, 0, 9, 12, Yes, Yes], [[a0, a1, a2, s], 9, 0, 9, 12, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (9, 9, [[[s], 9, 0, 9, 12, Yes, Yes], [[a0, s], 9, 0, 9, 12, Yes, Yes], [[a1, s], 9, 0, 9, 12, Yes, Yes], [[a0, a1, s], 9, 0, 9, 12, Yes, Yes], [[a2, s], 9, 0, 9, 12, Yes, Yes], [[a0, a2, s], 9, 0, 9, 12, Yes, Yes], [[a1, a2, s], 9, 0, 9, 12, Yes, Yes], [[a0, a1, a2, s], 9, 0, 9, 12, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0027/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 28. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(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)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0028/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 29. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10], candidate flows=[4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(30, 30, [[[s], 30, 0, 30, 35, Yes, Yes], [[a0, s], 30, 0, 30, 35, Yes, Yes], [[a1, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, s], 30, 0, 30, 35, Yes, Yes], [[a2, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, s], 30, 0, 30, 35, Yes, Yes], [[a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a3, s], 30, 0, 30, 35, Yes, Yes], [[a1, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a3, s], 30, 0, 30, 35, Yes, Yes], [[a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a4, s], 30, 0, 30, 35, Yes, Yes], [[a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (30, 30, [[[s], 30, 0, 30, 35, Yes, Yes], [[a0, s], 30, 0, 30, 35, Yes, Yes], [[a1, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, s], 30, 0, 30, 35, Yes, Yes], [[a2, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, s], 30, 0, 30, 35, Yes, Yes], [[a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a3, s], 30, 0, 30, 35, Yes, Yes], [[a1, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a3, s], 30, 0, 30, 35, Yes, Yes], [[a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a4, s], 30, 0, 30, 35, Yes, Yes], [[a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0029/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 30. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

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=[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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(11, 11, [[[s], 11, 0, 11, 13, Yes, Yes], [[a0, s], 11, 0, 11, 13, Yes, Yes], [[a1, s], 11, 0, 11, 13, Yes, Yes], [[a0, a1, s], 11, 0, 11, 13, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (11, 11, [[[s], 11, 0, 11, 13, Yes, Yes], [[a0, s], 11, 0, 11, 13, Yes, Yes], [[a1, s], 11, 0, 11, 13, Yes, Yes], [[a0, a1, s], 11, 0, 11, 13, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0030/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 31. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[7, 7, 8, 8, 9, 9], candidate flows=[6, 6, 7, 7, 8, 8]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(21, 21, [[[s], 21, 0, 21, 24, Yes, Yes], [[a0, s], 21, 0, 21, 24, Yes, Yes], [[a1, s], 21, 0, 21, 24, Yes, Yes], [[a0, a1, s], 21, 0, 21, 24, Yes, Yes], [[a2, s], 21, 0, 21, 24, Yes, Yes], [[a0, a2, s], 21, 0, 21, 24, Yes, Yes], [[a1, a2, s], 21, 0, 21, 24, Yes, Yes], [[a0, a1, a2, s], 21, 0, 21, 24, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (21, 21, [[[s], 21, 0, 21, 24, Yes, Yes], [[a0, s], 21, 0, 21, 24, Yes, Yes], [[a1, s], 21, 0, 21, 24, Yes, Yes], [[a0, a1, s], 21, 0, 21, 24, Yes, Yes], [[a2, s], 21, 0, 21, 24, Yes, Yes], [[a0, a2, s], 21, 0, 21, 24, Yes, Yes], [[a1, a2, s], 21, 0, 21, 24, Yes, Yes], [[a0, a1, a2, s], 21, 0, 21, 24, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0031/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 32. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

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=[8, 8, 9, 9, 10, 10, 11, 11, 13], candidate flows=[7, 7, 8, 8, 9, 9, 10, 10, 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(34, 34, [[[s], 34, 0, 34, 38, Yes, Yes], [[a0, s], 34, 0, 34, 38, Yes, Yes], [[a1, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, s], 34, 0, 34, 38, Yes, Yes], [[a2, s], 34, 0, 34, 38, Yes, Yes], [[a0, a2, s], 34, 0, 34, 38, Yes, Yes], [[a1, a2, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a2, s], 34, 0, 34, 38, Yes, Yes], [[a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a3, s], 34, 0, 34, 38, Yes, Yes], [[a1, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a3, s], 34, 0, 34, 38, Yes, Yes], [[a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a1, a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a2, a3, s], 34, 0, 34, 38, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (34, 34, [[[s], 34, 0, 34, 38, Yes, Yes], [[a0, s], 34, 0, 34, 38, Yes, Yes], [[a1, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, s], 34, 0, 34, 38, Yes, Yes], [[a2, s], 34, 0, 34, 38, Yes, Yes], [[a0, a2, s], 34, 0, 34, 38, Yes, Yes], [[a1, a2, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a2, s], 34, 0, 34, 38, Yes, Yes], [[a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a3, s], 34, 0, 34, 38, Yes, Yes], [[a1, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a3, s], 34, 0, 34, 38, Yes, Yes], [[a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a1, a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a2, a3, s], 34, 0, 34, 38, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0032/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 33. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13], candidate flows=[8, 8, 9, 9, 10, 10, 11, 11, 12, 12]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(50, 50, [[[s], 50, 0, 50, 55, Yes, Yes], [[a0, s], 50, 0, 50, 55, Yes, Yes], [[a1, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, s], 50, 0, 50, 55, Yes, Yes], [[a2, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, s], 50, 0, 50, 55, Yes, Yes], [[a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a3, s], 50, 0, 50, 55, Yes, Yes], [[a1, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a3, s], 50, 0, 50, 55, Yes, Yes], [[a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a4, s], 50, 0, 50, 55, Yes, Yes], [[a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (50, 50, [[[s], 50, 0, 50, 55, Yes, Yes], [[a0, s], 50, 0, 50, 55, Yes, Yes], [[a1, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, s], 50, 0, 50, 55, Yes, Yes], [[a2, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, s], 50, 0, 50, 55, Yes, Yes], [[a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a3, s], 50, 0, 50, 55, Yes, Yes], [[a1, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a3, s], 50, 0, 50, 55, Yes, Yes], [[a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a4, s], 50, 0, 50, 55, Yes, Yes], [[a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0033/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 34. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

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=[10, 10, 11, 11], candidate flows=[9, 9, 10, 10]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(19, 19, [[[s], 19, 0, 19, 21, Yes, Yes], [[a0, s], 19, 0, 19, 21, Yes, Yes], [[a1, s], 19, 0, 19, 21, Yes, Yes], [[a0, a1, s], 19, 0, 19, 21, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (19, 19, [[[s], 19, 0, 19, 21, Yes, Yes], [[a0, s], 19, 0, 19, 21, Yes, Yes], [[a1, s], 19, 0, 19, 21, Yes, Yes], [[a0, a1, s], 19, 0, 19, 21, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0034/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 35. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

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=[11, 11, 12, 12, 13, 13, 16], candidate flows=[10, 10, 11, 11, 12, 12, 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(33, 33, [[[s], 33, 0, 33, 36, Yes, Yes], [[a0, s], 33, 0, 33, 36, Yes, Yes], [[a1, s], 33, 0, 33, 36, Yes, Yes], [[a0, a1, s], 33, 0, 33, 36, Yes, Yes], [[a2, s], 33, 0, 33, 36, Yes, Yes], [[a0, a2, s], 33, 0, 33, 36, Yes, Yes], [[a1, a2, s], 33, 0, 33, 36, Yes, Yes], [[a0, a1, a2, s], 33, 0, 33, 36, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (33, 33, [[[s], 33, 0, 33, 36, Yes, Yes], [[a0, s], 33, 0, 33, 36, Yes, Yes], [[a1, s], 33, 0, 33, 36, Yes, Yes], [[a0, a1, s], 33, 0, 33, 36, Yes, Yes], [[a2, s], 33, 0, 33, 36, Yes, Yes], [[a0, a2, s], 33, 0, 33, 36, Yes, Yes], [[a1, a2, s], 33, 0, 33, 36, Yes, Yes], [[a0, a1, a2, s], 33, 0, 33, 36, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0035/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 36. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

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], candidate flows=[11, 11, 12, 12, 13, 13, 14, 14]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(50, 50, [[[s], 50, 0, 50, 54, Yes, Yes], [[a0, s], 50, 0, 50, 54, Yes, Yes], [[a1, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, s], 50, 0, 50, 54, Yes, Yes], [[a2, s], 50, 0, 50, 54, Yes, Yes], [[a0, a2, s], 50, 0, 50, 54, Yes, Yes], [[a1, a2, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a2, s], 50, 0, 50, 54, Yes, Yes], [[a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a3, s], 50, 0, 50, 54, Yes, Yes], [[a1, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a3, s], 50, 0, 50, 54, Yes, Yes], [[a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a1, a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a2, a3, s], 50, 0, 50, 54, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (50, 50, [[[s], 50, 0, 50, 54, Yes, Yes], [[a0, s], 50, 0, 50, 54, Yes, Yes], [[a1, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, s], 50, 0, 50, 54, Yes, Yes], [[a2, s], 50, 0, 50, 54, Yes, Yes], [[a0, a2, s], 50, 0, 50, 54, Yes, Yes], [[a1, a2, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a2, s], 50, 0, 50, 54, Yes, Yes], [[a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a3, s], 50, 0, 50, 54, Yes, Yes], [[a1, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a3, s], 50, 0, 50, 54, Yes, Yes], [[a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a1, a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a2, a3, s], 50, 0, 50, 54, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0036/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 37. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17], candidate flows=[12, 12, 13, 13, 14, 14, 15, 15, 16, 16]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(70, 70, [[[s], 70, 0, 70, 75, Yes, Yes], [[a0, s], 70, 0, 70, 75, Yes, Yes], [[a1, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, s], 70, 0, 70, 75, Yes, Yes], [[a2, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, s], 70, 0, 70, 75, Yes, Yes], [[a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a3, s], 70, 0, 70, 75, Yes, Yes], [[a1, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a3, s], 70, 0, 70, 75, Yes, Yes], [[a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a4, s], 70, 0, 70, 75, Yes, Yes], [[a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (70, 70, [[[s], 70, 0, 70, 75, Yes, Yes], [[a0, s], 70, 0, 70, 75, Yes, Yes], [[a1, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, s], 70, 0, 70, 75, Yes, Yes], [[a2, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, s], 70, 0, 70, 75, Yes, Yes], [[a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a3, s], 70, 0, 70, 75, Yes, Yes], [[a1, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a3, s], 70, 0, 70, 75, Yes, Yes], [[a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a4, s], 70, 0, 70, 75, Yes, Yes], [[a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0037/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 38. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[14, 14, 15, 15, 19], candidate flows=[13, 13, 14, 14, 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(27, 27, [[[s], 27, 0, 27, 29, Yes, Yes], [[a0, s], 27, 0, 27, 29, Yes, Yes], [[a1, s], 27, 0, 27, 29, Yes, Yes], [[a0, a1, s], 27, 0, 27, 29, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (27, 27, [[[s], 27, 0, 27, 29, Yes, Yes], [[a0, s], 27, 0, 27, 29, Yes, Yes], [[a1, s], 27, 0, 27, 29, Yes, Yes], [[a0, a1, s], 27, 0, 27, 29, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0038/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 39. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[15, 15, 16, 16, 17, 17], candidate flows=[14, 14, 15, 15, 16, 16]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(45, 45, [[[s], 45, 0, 45, 48, Yes, Yes], [[a0, s], 45, 0, 45, 48, Yes, Yes], [[a1, s], 45, 0, 45, 48, Yes, Yes], [[a0, a1, s], 45, 0, 45, 48, Yes, Yes], [[a2, s], 45, 0, 45, 48, Yes, Yes], [[a0, a2, s], 45, 0, 45, 48, Yes, Yes], [[a1, a2, s], 45, 0, 45, 48, Yes, Yes], [[a0, a1, a2, s], 45, 0, 45, 48, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (45, 45, [[[s], 45, 0, 45, 48, Yes, Yes], [[a0, s], 45, 0, 45, 48, Yes, Yes], [[a1, s], 45, 0, 45, 48, Yes, Yes], [[a0, a1, s], 45, 0, 45, 48, Yes, Yes], [[a2, s], 45, 0, 45, 48, Yes, Yes], [[a0, a2, s], 45, 0, 45, 48, Yes, Yes], [[a1, a2, s], 45, 0, 45, 48, Yes, Yes], [[a0, a1, a2, s], 45, 0, 45, 48, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0039/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 40. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

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=[16, 16, 17, 17, 18, 18, 19, 19], candidate flows=[15, 15, 16, 16, 17, 17, 18, 18]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(66, 66, [[[s], 66, 0, 66, 70, Yes, Yes], [[a0, s], 66, 0, 66, 70, Yes, Yes], [[a1, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, s], 66, 0, 66, 70, Yes, Yes], [[a2, s], 66, 0, 66, 70, Yes, Yes], [[a0, a2, s], 66, 0, 66, 70, Yes, Yes], [[a1, a2, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a2, s], 66, 0, 66, 70, Yes, Yes], [[a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a3, s], 66, 0, 66, 70, Yes, Yes], [[a1, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a3, s], 66, 0, 66, 70, Yes, Yes], [[a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a1, a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a2, a3, s], 66, 0, 66, 70, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (66, 66, [[[s], 66, 0, 66, 70, Yes, Yes], [[a0, s], 66, 0, 66, 70, Yes, Yes], [[a1, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, s], 66, 0, 66, 70, Yes, Yes], [[a2, s], 66, 0, 66, 70, Yes, Yes], [[a0, a2, s], 66, 0, 66, 70, Yes, Yes], [[a1, a2, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a2, s], 66, 0, 66, 70, Yes, Yes], [[a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a3, s], 66, 0, 66, 70, Yes, Yes], [[a1, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a3, s], 66, 0, 66, 70, Yes, Yes], [[a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a1, a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a2, a3, s], 66, 0, 66, 70, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0040/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 41. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22], candidate flows=[16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(90, 90, [[[s], 90, 0, 90, 95, Yes, Yes], [[a0, s], 90, 0, 90, 95, Yes, Yes], [[a1, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, s], 90, 0, 90, 95, Yes, Yes], [[a2, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, s], 90, 0, 90, 95, Yes, Yes], [[a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a3, s], 90, 0, 90, 95, Yes, Yes], [[a1, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a3, s], 90, 0, 90, 95, Yes, Yes], [[a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a4, s], 90, 0, 90, 95, Yes, Yes], [[a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (90, 90, [[[s], 90, 0, 90, 95, Yes, Yes], [[a0, s], 90, 0, 90, 95, Yes, Yes], [[a1, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, s], 90, 0, 90, 95, Yes, Yes], [[a2, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, s], 90, 0, 90, 95, Yes, Yes], [[a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a3, s], 90, 0, 90, 95, Yes, Yes], [[a1, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a3, s], 90, 0, 90, 95, Yes, Yes], [[a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a4, s], 90, 0, 90, 95, Yes, Yes], [[a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0041/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 42. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

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=[18, 18, 19, 19], candidate flows=[17, 17, 18, 18]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(35, 35, [[[s], 35, 0, 35, 37, Yes, Yes], [[a0, s], 35, 0, 35, 37, Yes, Yes], [[a1, s], 35, 0, 35, 37, Yes, Yes], [[a0, a1, s], 35, 0, 35, 37, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (35, 35, [[[s], 35, 0, 35, 37, Yes, Yes], [[a0, s], 35, 0, 35, 37, Yes, Yes], [[a1, s], 35, 0, 35, 37, Yes, Yes], [[a0, a1, s], 35, 0, 35, 37, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0042/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 43. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[19, 19, 20, 20, 21, 21], candidate flows=[18, 18, 19, 19, 20, 20]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(57, 57, [[[s], 57, 0, 57, 60, Yes, Yes], [[a0, s], 57, 0, 57, 60, Yes, Yes], [[a1, s], 57, 0, 57, 60, Yes, Yes], [[a0, a1, s], 57, 0, 57, 60, Yes, Yes], [[a2, s], 57, 0, 57, 60, Yes, Yes], [[a0, a2, s], 57, 0, 57, 60, Yes, Yes], [[a1, a2, s], 57, 0, 57, 60, Yes, Yes], [[a0, a1, a2, s], 57, 0, 57, 60, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (57, 57, [[[s], 57, 0, 57, 60, Yes, Yes], [[a0, s], 57, 0, 57, 60, Yes, Yes], [[a1, s], 57, 0, 57, 60, Yes, Yes], [[a0, a1, s], 57, 0, 57, 60, Yes, Yes], [[a2, s], 57, 0, 57, 60, Yes, Yes], [[a0, a2, s], 57, 0, 57, 60, Yes, Yes], [[a1, a2, s], 57, 0, 57, 60, Yes, Yes], [[a0, a1, a2, s], 57, 0, 57, 60, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0043/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 44. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(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)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0044/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 45. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25], candidate flows=[20, 20, 21, 21, 22, 22, 23, 23, 24, 24]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(110, 110, [[[s], 110, 0, 110, 115, Yes, Yes], [[a0, s], 110, 0, 110, 115, Yes, Yes], [[a1, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, s], 110, 0, 110, 115, Yes, Yes], [[a2, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, s], 110, 0, 110, 115, Yes, Yes], [[a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a3, s], 110, 0, 110, 115, Yes, Yes], [[a1, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a3, s], 110, 0, 110, 115, Yes, Yes], [[a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a4, s], 110, 0, 110, 115, Yes, Yes], [[a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (110, 110, [[[s], 110, 0, 110, 115, Yes, Yes], [[a0, s], 110, 0, 110, 115, Yes, Yes], [[a1, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, s], 110, 0, 110, 115, Yes, Yes], [[a2, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, s], 110, 0, 110, 115, Yes, Yes], [[a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a3, s], 110, 0, 110, 115, Yes, Yes], [[a1, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a3, s], 110, 0, 110, 115, Yes, Yes], [[a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a4, s], 110, 0, 110, 115, Yes, Yes], [[a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0045/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 46. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

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=[22, 22, 23, 23], candidate flows=[21, 21, 22, 22]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(43, 43, [[[s], 43, 0, 43, 45, Yes, Yes], [[a0, s], 43, 0, 43, 45, Yes, Yes], [[a1, s], 43, 0, 43, 45, Yes, Yes], [[a0, a1, s], 43, 0, 43, 45, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (43, 43, [[[s], 43, 0, 43, 45, Yes, Yes], [[a0, s], 43, 0, 43, 45, Yes, Yes], [[a1, s], 43, 0, 43, 45, Yes, Yes], [[a0, a1, s], 43, 0, 43, 45, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0046/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 47. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

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], candidate flows=[22, 22, 23, 23, 24, 24, 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(69, 69, [[[s], 69, 0, 69, 72, Yes, Yes], [[a0, s], 69, 0, 69, 72, Yes, Yes], [[a1, s], 69, 0, 69, 72, Yes, Yes], [[a0, a1, s], 69, 0, 69, 72, Yes, Yes], [[a2, s], 69, 0, 69, 72, Yes, Yes], [[a0, a2, s], 69, 0, 69, 72, Yes, Yes], [[a1, a2, s], 69, 0, 69, 72, Yes, Yes], [[a0, a1, a2, s], 69, 0, 69, 72, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (69, 69, [[[s], 69, 0, 69, 72, Yes, Yes], [[a0, s], 69, 0, 69, 72, Yes, Yes], [[a1, s], 69, 0, 69, 72, Yes, Yes], [[a0, a1, s], 69, 0, 69, 72, Yes, Yes], [[a2, s], 69, 0, 69, 72, Yes, Yes], [[a0, a2, s], 69, 0, 69, 72, Yes, Yes], [[a1, a2, s], 69, 0, 69, 72, Yes, Yes], [[a0, a1, a2, s], 69, 0, 69, 72, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0047/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 48. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

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=[24, 24, 25, 25, 26, 26, 27, 27], candidate flows=[23, 23, 24, 24, 25, 25, 26, 26]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(98, 98, [[[s], 98, 0, 98, 102, Yes, Yes], [[a0, s], 98, 0, 98, 102, Yes, Yes], [[a1, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, s], 98, 0, 98, 102, Yes, Yes], [[a2, s], 98, 0, 98, 102, Yes, Yes], [[a0, a2, s], 98, 0, 98, 102, Yes, Yes], [[a1, a2, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a2, s], 98, 0, 98, 102, Yes, Yes], [[a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a3, s], 98, 0, 98, 102, Yes, Yes], [[a1, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a3, s], 98, 0, 98, 102, Yes, Yes], [[a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a1, a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a2, a3, s], 98, 0, 98, 102, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (98, 98, [[[s], 98, 0, 98, 102, Yes, Yes], [[a0, s], 98, 0, 98, 102, Yes, Yes], [[a1, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, s], 98, 0, 98, 102, Yes, Yes], [[a2, s], 98, 0, 98, 102, Yes, Yes], [[a0, a2, s], 98, 0, 98, 102, Yes, Yes], [[a1, a2, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a2, s], 98, 0, 98, 102, Yes, Yes], [[a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a3, s], 98, 0, 98, 102, Yes, Yes], [[a1, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a3, s], 98, 0, 98, 102, Yes, Yes], [[a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a1, a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a2, a3, s], 98, 0, 98, 102, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0048/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 49. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29], candidate flows=[24, 24, 25, 25, 26, 26, 27, 27, 28, 28]. 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(130, 130, [[[s], 130, 0, 130, 135, Yes, Yes], [[a0, s], 130, 0, 130, 135, Yes, Yes], [[a1, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, s], 130, 0, 130, 135, Yes, Yes], [[a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (130, 130, [[[s], 130, 0, 130, 135, Yes, Yes], [[a0, s], 130, 0, 130, 135, Yes, Yes], [[a1, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, s], 130, 0, 130, 135, Yes, Yes], [[a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0049/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 50. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum

Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[26, 26, 27, 27, 31], candidate flows=[25, 25, 26, 26, 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).

Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.

View answer and explanation(51, 51, [[[s], 51, 0, 51, 53, Yes, Yes], [[a0, s], 51, 0, 51, 53, Yes, Yes], [[a1, s], 51, 0, 51, 53, Yes, Yes], [[a0, a1, s], 51, 0, 51, 53, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (51, 51, [[[s], 51, 0, 51, 53, Yes, Yes], [[a0, s], 51, 0, 51, 53, Yes, Yes], [[a1, s], 51, 0, 51, 53, Yes, Yes], [[a0, a1, s], 51, 0, 51, 53, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0050/)

Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 51. The Minimum Cut Is Selected from the Complete Cut Family

Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[2, 2, 3, 3, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 2], [[s, a1], 3]], 5], [[a0, s], [[[a0, t], 2], [[s, a1], 3]], 5], [[a1, s], [[[a1, t], 3], [[s, a0], 2]], 5], [[a0, a1, s], [[[a0, t], 2], [[a1, t], 3]], 5]], 5, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 2], [[s, a1], 3]], 5], [[a0, s], [[[a0, t], 2], [[s, a1], 3]], 5], [[a1, s], [[[a1, t], 3], [[s, a0], 2]], 5], [[a0, a1, s], [[[a0, t], 2], [[a1, t], 3]], 5]], 5, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0051/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 52. The Minimum Cut Is Selected from the Complete Cut Family

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[3, 3, 4, 4, 5, 5]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a0, s], [[[a0, t], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a1, s], [[[a1, t], 4], [[s, a0], 3], [[s, a2], 5]], 12], [[a0, a1, s], [[[a0, t], 3], [[a1, t], 4], [[s, a2], 5]], 12], [[a2, s], [[[a2, t], 5], [[s, a0], 3], [[s, a1], 4]], 12], [[a0, a2, s], [[[a0, t], 3], [[a2, t], 5], [[s, a1], 4]], 12], [[a1, a2, s], [[[a1, t], 4], [[a2, t], 5], [[s, a0], 3]], 12], [[a0, a1, a2, s], [[[a0, t], 3], [[a1, t], 4], [[a2, t], 5]], 12]], 12, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a0, s], [[[a0, t], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a1, s], [[[a1, t], 4], [[s, a0], 3], [[s, a2], 5]], 12], [[a0, a1, s], [[[a0, t], 3], [[a1, t], 4], [[s, a2], 5]], 12], [[a2, s], [[[a2, t], 5], [[s, a0], 3], [[s, a1], 4]], 12], [[a0, a2, s], [[[a0, t], 3], [[a2, t], 5], [[s, a1], 4]], 12], [[a1, a2, s], [[[a1, t], 4], [[a2, t], 5], [[s, a0], 3]], 12], [[a0, a1, a2, s], [[[a0, t], 3], [[a1, t], 4], [[a2, t], 5]], 12]], 12, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0052/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 53. The Minimum Cut Is Selected from the Complete Cut Family

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]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, s], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a1, s], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, a1, s], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a2, s], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22], [[a0, a2, s], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22], [[a1, a2, s], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22], [[a0, a1, a2, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22], [[a3, s], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22], [[a0, a3, s], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22], [[a1, a3, s], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22], [[a0, a1, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22], [[a2, a3, s], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22], [[a0, a2, a3, s], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22], [[a1, a2, a3, s], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22], [[a0, a1, a2, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]], 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]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, s], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a1, s], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, a1, s], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a2, s], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22], [[a0, a2, s], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22], [[a1, a2, s], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22], [[a0, a1, a2, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22], [[a3, s], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22], [[a0, a3, s], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22], [[a1, a3, s], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22], [[a0, a1, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22], [[a2, a3, s], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22], [[a0, a2, a3, s], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22], [[a1, a2, a3, s], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22], [[a0, a1, a2, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]], 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]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0053/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 54. The Minimum Cut Is Selected from the Complete Cut Family

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, s], [[[a0, t], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, s], [[[a1, t], 6], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, s], [[[a0, t], 5], [[a1, t], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a2, s], [[[a2, t], 7], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a2, s], [[[a0, t], 5], [[a2, t], 7], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, a2, s], [[[a1, t], 6], [[a2, t], 7], [[s, a0], 5], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, a2, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a3, s], [[[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a3, s], [[[a0, t], 5], [[a3, t], 8], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a1, a3, s], [[[a1, t], 6], [[a3, t], 8], [[s, a0], 5], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a1, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[s, a2], 7], [[s, a4], 9]], 35], [[a2, a3, s], [[[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a4], 9]], 35], [[a0, a2, a3, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[s, a1], 6], [[s, a4], 9]], 35], [[a1, a2, a3, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a4], 9]], 35], [[a0, a1, a2, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a4], 9]], 35], [[a4, s], [[[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a4, s], [[[a0, t], 5], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a1, a4, s], [[[a1, t], 6], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a1, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a4, t], 9], [[s, a2], 7], [[s, a3], 8]], 35], [[a2, a4, s], [[[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8]], 35], [[a0, a2, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a4, t], 9], [[s, a1], 6], [[s, a3], 8]], 35], [[a1, a2, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a3], 8]], 35], [[a0, a1, a2, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a3], 8]], 35], [[a3, a4, s], [[[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7]], 35], [[a0, a3, a4, s], [[[a0, t], 5], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7]], 35], [[a1, a3, a4, s], [[[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7]], 35], [[a0, a1, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a2], 7]], 35], [[a2, a3, a4, s], [[[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6]], 35], [[a0, a2, a3, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6]], 35], [[a1, a2, a3, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5]], 35], [[a0, a1, a2, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9]], 35]], 35, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, s], [[[a0, t], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, s], [[[a1, t], 6], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, s], [[[a0, t], 5], [[a1, t], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a2, s], [[[a2, t], 7], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a2, s], [[[a0, t], 5], [[a2, t], 7], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, a2, s], [[[a1, t], 6], [[a2, t], 7], [[s, a0], 5], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, a2, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a3, s], [[[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a3, s], [[[a0, t], 5], [[a3, t], 8], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a1, a3, s], [[[a1, t], 6], [[a3, t], 8], [[s, a0], 5], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a1, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[s, a2], 7], [[s, a4], 9]], 35], [[a2, a3, s], [[[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a4], 9]], 35], [[a0, a2, a3, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[s, a1], 6], [[s, a4], 9]], 35], [[a1, a2, a3, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a4], 9]], 35], [[a0, a1, a2, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a4], 9]], 35], [[a4, s], [[[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a4, s], [[[a0, t], 5], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a1, a4, s], [[[a1, t], 6], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a1, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a4, t], 9], [[s, a2], 7], [[s, a3], 8]], 35], [[a2, a4, s], [[[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8]], 35], [[a0, a2, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a4, t], 9], [[s, a1], 6], [[s, a3], 8]], 35], [[a1, a2, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a3], 8]], 35], [[a0, a1, a2, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a3], 8]], 35], [[a3, a4, s], [[[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7]], 35], [[a0, a3, a4, s], [[[a0, t], 5], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7]], 35], [[a1, a3, a4, s], [[[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7]], 35], [[a0, a1, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a2], 7]], 35], [[a2, a3, a4, s], [[[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6]], 35], [[a0, a2, a3, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6]], 35], [[a1, a2, a3, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5]], 35], [[a0, a1, a2, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9]], 35]], 35, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0054/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 55. The Minimum Cut Is Selected from the Complete Cut Family

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]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 6], [[s, a1], 7]], 13], [[a0, s], [[[a0, t], 6], [[s, a1], 7]], 13], [[a1, s], [[[a1, t], 7], [[s, a0], 6]], 13], [[a0, a1, s], [[[a0, t], 6], [[a1, t], 7]], 13]], 13, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 6], [[s, a1], 7]], 13], [[a0, s], [[[a0, t], 6], [[s, a1], 7]], 13], [[a1, s], [[[a1, t], 7], [[s, a0], 6]], 13], [[a0, a1, s], [[[a0, t], 6], [[a1, t], 7]], 13]], 13, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0055/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 56. The Minimum Cut Is Selected from the Complete Cut Family

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[7, 7, 8, 8, 9, 9]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a0, s], [[[a0, t], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a1, s], [[[a1, t], 8], [[s, a0], 7], [[s, a2], 9]], 24], [[a0, a1, s], [[[a0, t], 7], [[a1, t], 8], [[s, a2], 9]], 24], [[a2, s], [[[a2, t], 9], [[s, a0], 7], [[s, a1], 8]], 24], [[a0, a2, s], [[[a0, t], 7], [[a2, t], 9], [[s, a1], 8]], 24], [[a1, a2, s], [[[a1, t], 8], [[a2, t], 9], [[s, a0], 7]], 24], [[a0, a1, a2, s], [[[a0, t], 7], [[a1, t], 8], [[a2, t], 9]], 24]], 24, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a0, s], [[[a0, t], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a1, s], [[[a1, t], 8], [[s, a0], 7], [[s, a2], 9]], 24], [[a0, a1, s], [[[a0, t], 7], [[a1, t], 8], [[s, a2], 9]], 24], [[a2, s], [[[a2, t], 9], [[s, a0], 7], [[s, a1], 8]], 24], [[a0, a2, s], [[[a0, t], 7], [[a2, t], 9], [[s, a1], 8]], 24], [[a1, a2, s], [[[a1, t], 8], [[a2, t], 9], [[s, a0], 7]], 24], [[a0, a1, a2, s], [[[a0, t], 7], [[a1, t], 8], [[a2, t], 9]], 24]], 24, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0056/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 57. The Minimum Cut Is Selected from the Complete Cut Family

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=[8, 8, 9, 9, 10, 10, 11, 11, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, s], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a1, s], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, a1, s], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a2, s], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38], [[a0, a2, s], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38], [[a1, a2, s], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38], [[a0, a1, a2, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38], [[a3, s], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38], [[a0, a3, s], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38], [[a1, a3, s], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38], [[a0, a1, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38], [[a2, a3, s], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38], [[a0, a2, a3, s], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38], [[a1, a2, a3, s], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38], [[a0, a1, a2, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]], 38, [[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]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, s], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a1, s], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, a1, s], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a2, s], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38], [[a0, a2, s], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38], [[a1, a2, s], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38], [[a0, a1, a2, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38], [[a3, s], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38], [[a0, a3, s], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38], [[a1, a3, s], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38], [[a0, a1, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38], [[a2, a3, s], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38], [[a0, a2, a3, s], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38], [[a1, a2, a3, s], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38], [[a0, a1, a2, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]], 38, [[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]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0057/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 58. The Minimum Cut Is Selected from the Complete Cut Family

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 55, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 55, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0058/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 59. The Minimum Cut Is Selected from the Complete Cut Family

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=[10, 10, 11, 11]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 10], [[s, a1], 11]], 21], [[a0, s], [[[a0, t], 10], [[s, a1], 11]], 21], [[a1, s], [[[a1, t], 11], [[s, a0], 10]], 21], [[a0, a1, s], [[[a0, t], 10], [[a1, t], 11]], 21]], 21, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 10], [[s, a1], 11]], 21], [[a0, s], [[[a0, t], 10], [[s, a1], 11]], 21], [[a1, s], [[[a1, t], 11], [[s, a0], 10]], 21], [[a0, a1, s], [[[a0, t], 10], [[a1, t], 11]], 21]], 21, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0059/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 60. The Minimum Cut Is Selected from the Complete Cut Family

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=[11, 11, 12, 12, 13, 13, 16]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a0, s], [[[a0, t], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a1, s], [[[a1, t], 12], [[s, a0], 11], [[s, a2], 13]], 36], [[a0, a1, s], [[[a0, t], 11], [[a1, t], 12], [[s, a2], 13]], 36], [[a2, s], [[[a2, t], 13], [[s, a0], 11], [[s, a1], 12]], 36], [[a0, a2, s], [[[a0, t], 11], [[a2, t], 13], [[s, a1], 12]], 36], [[a1, a2, s], [[[a1, t], 12], [[a2, t], 13], [[s, a0], 11]], 36], [[a0, a1, a2, s], [[[a0, t], 11], [[a1, t], 12], [[a2, t], 13]], 36]], 36, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a0, s], [[[a0, t], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a1, s], [[[a1, t], 12], [[s, a0], 11], [[s, a2], 13]], 36], [[a0, a1, s], [[[a0, t], 11], [[a1, t], 12], [[s, a2], 13]], 36], [[a2, s], [[[a2, t], 13], [[s, a0], 11], [[s, a1], 12]], 36], [[a0, a2, s], [[[a0, t], 11], [[a2, t], 13], [[s, a1], 12]], 36], [[a1, a2, s], [[[a1, t], 12], [[a2, t], 13], [[s, a0], 11]], 36], [[a0, a1, a2, s], [[[a0, t], 11], [[a1, t], 12], [[a2, t], 13]], 36]], 36, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0060/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 61. The Minimum Cut Is Selected from the Complete Cut Family

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 (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[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]], 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]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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]], 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]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0061/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 62. The Minimum Cut Is Selected from the Complete Cut Family

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, s], [[[a0, t], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, s], [[[a1, t], 14], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, s], [[[a0, t], 13], [[a1, t], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a2, s], [[[a2, t], 15], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a2, s], [[[a0, t], 13], [[a2, t], 15], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, a2, s], [[[a1, t], 14], [[a2, t], 15], [[s, a0], 13], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, a2, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a3, s], [[[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a3, s], [[[a0, t], 13], [[a3, t], 16], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a1, a3, s], [[[a1, t], 14], [[a3, t], 16], [[s, a0], 13], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a1, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[s, a2], 15], [[s, a4], 17]], 75], [[a2, a3, s], [[[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a4], 17]], 75], [[a0, a2, a3, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[s, a1], 14], [[s, a4], 17]], 75], [[a1, a2, a3, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a4], 17]], 75], [[a0, a1, a2, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a4], 17]], 75], [[a4, s], [[[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a4, s], [[[a0, t], 13], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a1, a4, s], [[[a1, t], 14], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a1, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a4, t], 17], [[s, a2], 15], [[s, a3], 16]], 75], [[a2, a4, s], [[[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16]], 75], [[a0, a2, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a4, t], 17], [[s, a1], 14], [[s, a3], 16]], 75], [[a1, a2, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a3], 16]], 75], [[a0, a1, a2, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a3], 16]], 75], [[a3, a4, s], [[[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15]], 75], [[a0, a3, a4, s], [[[a0, t], 13], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15]], 75], [[a1, a3, a4, s], [[[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15]], 75], [[a0, a1, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a2], 15]], 75], [[a2, a3, a4, s], [[[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14]], 75], [[a0, a2, a3, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14]], 75], [[a1, a2, a3, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13]], 75], [[a0, a1, a2, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17]], 75]], 75, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, s], [[[a0, t], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, s], [[[a1, t], 14], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, s], [[[a0, t], 13], [[a1, t], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a2, s], [[[a2, t], 15], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a2, s], [[[a0, t], 13], [[a2, t], 15], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, a2, s], [[[a1, t], 14], [[a2, t], 15], [[s, a0], 13], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, a2, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a3, s], [[[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a3, s], [[[a0, t], 13], [[a3, t], 16], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a1, a3, s], [[[a1, t], 14], [[a3, t], 16], [[s, a0], 13], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a1, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[s, a2], 15], [[s, a4], 17]], 75], [[a2, a3, s], [[[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a4], 17]], 75], [[a0, a2, a3, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[s, a1], 14], [[s, a4], 17]], 75], [[a1, a2, a3, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a4], 17]], 75], [[a0, a1, a2, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a4], 17]], 75], [[a4, s], [[[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a4, s], [[[a0, t], 13], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a1, a4, s], [[[a1, t], 14], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a1, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a4, t], 17], [[s, a2], 15], [[s, a3], 16]], 75], [[a2, a4, s], [[[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16]], 75], [[a0, a2, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a4, t], 17], [[s, a1], 14], [[s, a3], 16]], 75], [[a1, a2, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a3], 16]], 75], [[a0, a1, a2, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a3], 16]], 75], [[a3, a4, s], [[[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15]], 75], [[a0, a3, a4, s], [[[a0, t], 13], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15]], 75], [[a1, a3, a4, s], [[[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15]], 75], [[a0, a1, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a2], 15]], 75], [[a2, a3, a4, s], [[[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14]], 75], [[a0, a2, a3, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14]], 75], [[a1, a2, a3, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13]], 75], [[a0, a1, a2, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17]], 75]], 75, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0062/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 63. The Minimum Cut Is Selected from the Complete Cut Family

Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[14, 14, 15, 15, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 14], [[s, a1], 15]], 29], [[a0, s], [[[a0, t], 14], [[s, a1], 15]], 29], [[a1, s], [[[a1, t], 15], [[s, a0], 14]], 29], [[a0, a1, s], [[[a0, t], 14], [[a1, t], 15]], 29]], 29, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 14], [[s, a1], 15]], 29], [[a0, s], [[[a0, t], 14], [[s, a1], 15]], 29], [[a1, s], [[[a1, t], 15], [[s, a0], 14]], 29], [[a0, a1, s], [[[a0, t], 14], [[a1, t], 15]], 29]], 29, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0063/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 64. The Minimum Cut Is Selected from the Complete Cut Family

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a0, s], [[[a0, t], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a1, s], [[[a1, t], 16], [[s, a0], 15], [[s, a2], 17]], 48], [[a0, a1, s], [[[a0, t], 15], [[a1, t], 16], [[s, a2], 17]], 48], [[a2, s], [[[a2, t], 17], [[s, a0], 15], [[s, a1], 16]], 48], [[a0, a2, s], [[[a0, t], 15], [[a2, t], 17], [[s, a1], 16]], 48], [[a1, a2, s], [[[a1, t], 16], [[a2, t], 17], [[s, a0], 15]], 48], [[a0, a1, a2, s], [[[a0, t], 15], [[a1, t], 16], [[a2, t], 17]], 48]], 48, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a0, s], [[[a0, t], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a1, s], [[[a1, t], 16], [[s, a0], 15], [[s, a2], 17]], 48], [[a0, a1, s], [[[a0, t], 15], [[a1, t], 16], [[s, a2], 17]], 48], [[a2, s], [[[a2, t], 17], [[s, a0], 15], [[s, a1], 16]], 48], [[a0, a2, s], [[[a0, t], 15], [[a2, t], 17], [[s, a1], 16]], 48], [[a1, a2, s], [[[a1, t], 16], [[a2, t], 17], [[s, a0], 15]], 48], [[a0, a1, a2, s], [[[a0, t], 15], [[a1, t], 16], [[a2, t], 17]], 48]], 48, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0064/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 65. The Minimum Cut Is Selected from the Complete Cut Family

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=[16, 16, 17, 17, 18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, s], [[[a0, t], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a1, s], [[[a1, t], 17], [[s, a0], 16], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, a1, s], [[[a0, t], 16], [[a1, t], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a2, s], [[[a2, t], 18], [[s, a0], 16], [[s, a1], 17], [[s, a3], 19]], 70], [[a0, a2, s], [[[a0, t], 16], [[a2, t], 18], [[s, a1], 17], [[s, a3], 19]], 70], [[a1, a2, s], [[[a1, t], 17], [[a2, t], 18], [[s, a0], 16], [[s, a3], 19]], 70], [[a0, a1, a2, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[s, a3], 19]], 70], [[a3, s], [[[a3, t], 19], [[s, a0], 16], [[s, a1], 17], [[s, a2], 18]], 70], [[a0, a3, s], [[[a0, t], 16], [[a3, t], 19], [[s, a1], 17], [[s, a2], 18]], 70], [[a1, a3, s], [[[a1, t], 17], [[a3, t], 19], [[s, a0], 16], [[s, a2], 18]], 70], [[a0, a1, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a3, t], 19], [[s, a2], 18]], 70], [[a2, a3, s], [[[a2, t], 18], [[a3, t], 19], [[s, a0], 16], [[s, a1], 17]], 70], [[a0, a2, a3, s], [[[a0, t], 16], [[a2, t], 18], [[a3, t], 19], [[s, a1], 17]], 70], [[a1, a2, a3, s], [[[a1, t], 17], [[a2, t], 18], [[a3, t], 19], [[s, a0], 16]], 70], [[a0, a1, a2, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[a3, t], 19]], 70]], 70, [[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]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, s], [[[a0, t], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a1, s], [[[a1, t], 17], [[s, a0], 16], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, a1, s], [[[a0, t], 16], [[a1, t], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a2, s], [[[a2, t], 18], [[s, a0], 16], [[s, a1], 17], [[s, a3], 19]], 70], [[a0, a2, s], [[[a0, t], 16], [[a2, t], 18], [[s, a1], 17], [[s, a3], 19]], 70], [[a1, a2, s], [[[a1, t], 17], [[a2, t], 18], [[s, a0], 16], [[s, a3], 19]], 70], [[a0, a1, a2, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[s, a3], 19]], 70], [[a3, s], [[[a3, t], 19], [[s, a0], 16], [[s, a1], 17], [[s, a2], 18]], 70], [[a0, a3, s], [[[a0, t], 16], [[a3, t], 19], [[s, a1], 17], [[s, a2], 18]], 70], [[a1, a3, s], [[[a1, t], 17], [[a3, t], 19], [[s, a0], 16], [[s, a2], 18]], 70], [[a0, a1, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a3, t], 19], [[s, a2], 18]], 70], [[a2, a3, s], [[[a2, t], 18], [[a3, t], 19], [[s, a0], 16], [[s, a1], 17]], 70], [[a0, a2, a3, s], [[[a0, t], 16], [[a2, t], 18], [[a3, t], 19], [[s, a1], 17]], 70], [[a1, a2, a3, s], [[[a1, t], 17], [[a2, t], 18], [[a3, t], 19], [[s, a0], 16]], 70], [[a0, a1, a2, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[a3, t], 19]], 70]], 70, [[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]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0065/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 66. The Minimum Cut Is Selected from the Complete Cut Family

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, s], [[[a0, t], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, s], [[[a1, t], 18], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, s], [[[a0, t], 17], [[a1, t], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a2, s], [[[a2, t], 19], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a2, s], [[[a0, t], 17], [[a2, t], 19], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, a2, s], [[[a1, t], 18], [[a2, t], 19], [[s, a0], 17], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, a2, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a3, s], [[[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a3, s], [[[a0, t], 17], [[a3, t], 20], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a1, a3, s], [[[a1, t], 18], [[a3, t], 20], [[s, a0], 17], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a1, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[s, a2], 19], [[s, a4], 21]], 95], [[a2, a3, s], [[[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a4], 21]], 95], [[a0, a2, a3, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[s, a1], 18], [[s, a4], 21]], 95], [[a1, a2, a3, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a4], 21]], 95], [[a0, a1, a2, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a4], 21]], 95], [[a4, s], [[[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a4, s], [[[a0, t], 17], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a1, a4, s], [[[a1, t], 18], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a1, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a4, t], 21], [[s, a2], 19], [[s, a3], 20]], 95], [[a2, a4, s], [[[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20]], 95], [[a0, a2, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a4, t], 21], [[s, a1], 18], [[s, a3], 20]], 95], [[a1, a2, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a3], 20]], 95], [[a0, a1, a2, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a3], 20]], 95], [[a3, a4, s], [[[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19]], 95], [[a0, a3, a4, s], [[[a0, t], 17], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19]], 95], [[a1, a3, a4, s], [[[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19]], 95], [[a0, a1, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a2], 19]], 95], [[a2, a3, a4, s], [[[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18]], 95], [[a0, a2, a3, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18]], 95], [[a1, a2, a3, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17]], 95], [[a0, a1, a2, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21]], 95]], 95, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, s], [[[a0, t], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, s], [[[a1, t], 18], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, s], [[[a0, t], 17], [[a1, t], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a2, s], [[[a2, t], 19], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a2, s], [[[a0, t], 17], [[a2, t], 19], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, a2, s], [[[a1, t], 18], [[a2, t], 19], [[s, a0], 17], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, a2, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a3, s], [[[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a3, s], [[[a0, t], 17], [[a3, t], 20], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a1, a3, s], [[[a1, t], 18], [[a3, t], 20], [[s, a0], 17], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a1, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[s, a2], 19], [[s, a4], 21]], 95], [[a2, a3, s], [[[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a4], 21]], 95], [[a0, a2, a3, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[s, a1], 18], [[s, a4], 21]], 95], [[a1, a2, a3, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a4], 21]], 95], [[a0, a1, a2, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a4], 21]], 95], [[a4, s], [[[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a4, s], [[[a0, t], 17], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a1, a4, s], [[[a1, t], 18], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a1, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a4, t], 21], [[s, a2], 19], [[s, a3], 20]], 95], [[a2, a4, s], [[[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20]], 95], [[a0, a2, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a4, t], 21], [[s, a1], 18], [[s, a3], 20]], 95], [[a1, a2, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a3], 20]], 95], [[a0, a1, a2, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a3], 20]], 95], [[a3, a4, s], [[[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19]], 95], [[a0, a3, a4, s], [[[a0, t], 17], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19]], 95], [[a1, a3, a4, s], [[[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19]], 95], [[a0, a1, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a2], 19]], 95], [[a2, a3, a4, s], [[[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18]], 95], [[a0, a2, a3, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18]], 95], [[a1, a2, a3, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17]], 95], [[a0, a1, a2, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21]], 95]], 95, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0066/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 67. The Minimum Cut Is Selected from the Complete Cut Family

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=[18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 18], [[s, a1], 19]], 37], [[a0, s], [[[a0, t], 18], [[s, a1], 19]], 37], [[a1, s], [[[a1, t], 19], [[s, a0], 18]], 37], [[a0, a1, s], [[[a0, t], 18], [[a1, t], 19]], 37]], 37, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 18], [[s, a1], 19]], 37], [[a0, s], [[[a0, t], 18], [[s, a1], 19]], 37], [[a1, s], [[[a1, t], 19], [[s, a0], 18]], 37], [[a0, a1, s], [[[a0, t], 18], [[a1, t], 19]], 37]], 37, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0067/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 68. The Minimum Cut Is Selected from the Complete Cut Family

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[19, 19, 20, 20, 21, 21]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a0, s], [[[a0, t], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a1, s], [[[a1, t], 20], [[s, a0], 19], [[s, a2], 21]], 60], [[a0, a1, s], [[[a0, t], 19], [[a1, t], 20], [[s, a2], 21]], 60], [[a2, s], [[[a2, t], 21], [[s, a0], 19], [[s, a1], 20]], 60], [[a0, a2, s], [[[a0, t], 19], [[a2, t], 21], [[s, a1], 20]], 60], [[a1, a2, s], [[[a1, t], 20], [[a2, t], 21], [[s, a0], 19]], 60], [[a0, a1, a2, s], [[[a0, t], 19], [[a1, t], 20], [[a2, t], 21]], 60]], 60, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a0, s], [[[a0, t], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a1, s], [[[a1, t], 20], [[s, a0], 19], [[s, a2], 21]], 60], [[a0, a1, s], [[[a0, t], 19], [[a1, t], 20], [[s, a2], 21]], 60], [[a2, s], [[[a2, t], 21], [[s, a0], 19], [[s, a1], 20]], 60], [[a0, a2, s], [[[a0, t], 19], [[a2, t], 21], [[s, a1], 20]], 60], [[a1, a2, s], [[[a1, t], 20], [[a2, t], 21], [[s, a0], 19]], 60], [[a0, a1, a2, s], [[[a0, t], 19], [[a1, t], 20], [[a2, t], 21]], 60]], 60, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0068/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 69. The Minimum Cut Is Selected from the Complete Cut Family

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]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, s], [[[a0, t], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a1, s], [[[a1, t], 21], [[s, a0], 20], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, a1, s], [[[a0, t], 20], [[a1, t], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a2, s], [[[a2, t], 22], [[s, a0], 20], [[s, a1], 21], [[s, a3], 23]], 86], [[a0, a2, s], [[[a0, t], 20], [[a2, t], 22], [[s, a1], 21], [[s, a3], 23]], 86], [[a1, a2, s], [[[a1, t], 21], [[a2, t], 22], [[s, a0], 20], [[s, a3], 23]], 86], [[a0, a1, a2, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[s, a3], 23]], 86], [[a3, s], [[[a3, t], 23], [[s, a0], 20], [[s, a1], 21], [[s, a2], 22]], 86], [[a0, a3, s], [[[a0, t], 20], [[a3, t], 23], [[s, a1], 21], [[s, a2], 22]], 86], [[a1, a3, s], [[[a1, t], 21], [[a3, t], 23], [[s, a0], 20], [[s, a2], 22]], 86], [[a0, a1, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a3, t], 23], [[s, a2], 22]], 86], [[a2, a3, s], [[[a2, t], 22], [[a3, t], 23], [[s, a0], 20], [[s, a1], 21]], 86], [[a0, a2, a3, s], [[[a0, t], 20], [[a2, t], 22], [[a3, t], 23], [[s, a1], 21]], 86], [[a1, a2, a3, s], [[[a1, t], 21], [[a2, t], 22], [[a3, t], 23], [[s, a0], 20]], 86], [[a0, a1, a2, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[a3, t], 23]], 86]], 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]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, s], [[[a0, t], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a1, s], [[[a1, t], 21], [[s, a0], 20], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, a1, s], [[[a0, t], 20], [[a1, t], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a2, s], [[[a2, t], 22], [[s, a0], 20], [[s, a1], 21], [[s, a3], 23]], 86], [[a0, a2, s], [[[a0, t], 20], [[a2, t], 22], [[s, a1], 21], [[s, a3], 23]], 86], [[a1, a2, s], [[[a1, t], 21], [[a2, t], 22], [[s, a0], 20], [[s, a3], 23]], 86], [[a0, a1, a2, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[s, a3], 23]], 86], [[a3, s], [[[a3, t], 23], [[s, a0], 20], [[s, a1], 21], [[s, a2], 22]], 86], [[a0, a3, s], [[[a0, t], 20], [[a3, t], 23], [[s, a1], 21], [[s, a2], 22]], 86], [[a1, a3, s], [[[a1, t], 21], [[a3, t], 23], [[s, a0], 20], [[s, a2], 22]], 86], [[a0, a1, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a3, t], 23], [[s, a2], 22]], 86], [[a2, a3, s], [[[a2, t], 22], [[a3, t], 23], [[s, a0], 20], [[s, a1], 21]], 86], [[a0, a2, a3, s], [[[a0, t], 20], [[a2, t], 22], [[a3, t], 23], [[s, a1], 21]], 86], [[a1, a2, a3, s], [[[a1, t], 21], [[a2, t], 22], [[a3, t], 23], [[s, a0], 20]], 86], [[a0, a1, a2, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[a3, t], 23]], 86]], 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]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0069/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 70. The Minimum Cut Is Selected from the Complete Cut Family

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, s], [[[a0, t], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, s], [[[a1, t], 22], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, s], [[[a0, t], 21], [[a1, t], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a2, s], [[[a2, t], 23], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a2, s], [[[a0, t], 21], [[a2, t], 23], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, a2, s], [[[a1, t], 22], [[a2, t], 23], [[s, a0], 21], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, a2, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a3, s], [[[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a3, s], [[[a0, t], 21], [[a3, t], 24], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a1, a3, s], [[[a1, t], 22], [[a3, t], 24], [[s, a0], 21], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a1, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[s, a2], 23], [[s, a4], 25]], 115], [[a2, a3, s], [[[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a4], 25]], 115], [[a0, a2, a3, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[s, a1], 22], [[s, a4], 25]], 115], [[a1, a2, a3, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a4], 25]], 115], [[a0, a1, a2, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a4], 25]], 115], [[a4, s], [[[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a4, s], [[[a0, t], 21], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a1, a4, s], [[[a1, t], 22], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a1, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a4, t], 25], [[s, a2], 23], [[s, a3], 24]], 115], [[a2, a4, s], [[[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24]], 115], [[a0, a2, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a4, t], 25], [[s, a1], 22], [[s, a3], 24]], 115], [[a1, a2, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a3], 24]], 115], [[a0, a1, a2, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a3], 24]], 115], [[a3, a4, s], [[[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23]], 115], [[a0, a3, a4, s], [[[a0, t], 21], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23]], 115], [[a1, a3, a4, s], [[[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23]], 115], [[a0, a1, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a2], 23]], 115], [[a2, a3, a4, s], [[[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22]], 115], [[a0, a2, a3, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22]], 115], [[a1, a2, a3, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21]], 115], [[a0, a1, a2, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25]], 115]], 115, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, s], [[[a0, t], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, s], [[[a1, t], 22], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, s], [[[a0, t], 21], [[a1, t], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a2, s], [[[a2, t], 23], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a2, s], [[[a0, t], 21], [[a2, t], 23], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, a2, s], [[[a1, t], 22], [[a2, t], 23], [[s, a0], 21], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, a2, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a3, s], [[[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a3, s], [[[a0, t], 21], [[a3, t], 24], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a1, a3, s], [[[a1, t], 22], [[a3, t], 24], [[s, a0], 21], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a1, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[s, a2], 23], [[s, a4], 25]], 115], [[a2, a3, s], [[[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a4], 25]], 115], [[a0, a2, a3, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[s, a1], 22], [[s, a4], 25]], 115], [[a1, a2, a3, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a4], 25]], 115], [[a0, a1, a2, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a4], 25]], 115], [[a4, s], [[[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a4, s], [[[a0, t], 21], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a1, a4, s], [[[a1, t], 22], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a1, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a4, t], 25], [[s, a2], 23], [[s, a3], 24]], 115], [[a2, a4, s], [[[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24]], 115], [[a0, a2, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a4, t], 25], [[s, a1], 22], [[s, a3], 24]], 115], [[a1, a2, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a3], 24]], 115], [[a0, a1, a2, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a3], 24]], 115], [[a3, a4, s], [[[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23]], 115], [[a0, a3, a4, s], [[[a0, t], 21], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23]], 115], [[a1, a3, a4, s], [[[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23]], 115], [[a0, a1, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a2], 23]], 115], [[a2, a3, a4, s], [[[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22]], 115], [[a0, a2, a3, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22]], 115], [[a1, a2, a3, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21]], 115], [[a0, a1, a2, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25]], 115]], 115, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0070/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 71. The Minimum Cut Is Selected from the Complete Cut Family

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=[22, 22, 23, 23]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 22], [[s, a1], 23]], 45], [[a0, s], [[[a0, t], 22], [[s, a1], 23]], 45], [[a1, s], [[[a1, t], 23], [[s, a0], 22]], 45], [[a0, a1, s], [[[a0, t], 22], [[a1, t], 23]], 45]], 45, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 22], [[s, a1], 23]], 45], [[a0, s], [[[a0, t], 22], [[s, a1], 23]], 45], [[a1, s], [[[a1, t], 23], [[s, a0], 22]], 45], [[a0, a1, s], [[[a0, t], 22], [[a1, t], 23]], 45]], 45, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0071/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 72. The Minimum Cut Is Selected from the Complete Cut Family

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 (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[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]], 72, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[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]], 72, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0072/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 73. The Minimum Cut Is Selected from the Complete Cut Family

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=[24, 24, 25, 25, 26, 26, 27, 27]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, s], [[[a0, t], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a1, s], [[[a1, t], 25], [[s, a0], 24], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, a1, s], [[[a0, t], 24], [[a1, t], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a2, s], [[[a2, t], 26], [[s, a0], 24], [[s, a1], 25], [[s, a3], 27]], 102], [[a0, a2, s], [[[a0, t], 24], [[a2, t], 26], [[s, a1], 25], [[s, a3], 27]], 102], [[a1, a2, s], [[[a1, t], 25], [[a2, t], 26], [[s, a0], 24], [[s, a3], 27]], 102], [[a0, a1, a2, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[s, a3], 27]], 102], [[a3, s], [[[a3, t], 27], [[s, a0], 24], [[s, a1], 25], [[s, a2], 26]], 102], [[a0, a3, s], [[[a0, t], 24], [[a3, t], 27], [[s, a1], 25], [[s, a2], 26]], 102], [[a1, a3, s], [[[a1, t], 25], [[a3, t], 27], [[s, a0], 24], [[s, a2], 26]], 102], [[a0, a1, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a3, t], 27], [[s, a2], 26]], 102], [[a2, a3, s], [[[a2, t], 26], [[a3, t], 27], [[s, a0], 24], [[s, a1], 25]], 102], [[a0, a2, a3, s], [[[a0, t], 24], [[a2, t], 26], [[a3, t], 27], [[s, a1], 25]], 102], [[a1, a2, a3, s], [[[a1, t], 25], [[a2, t], 26], [[a3, t], 27], [[s, a0], 24]], 102], [[a0, a1, a2, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[a3, t], 27]], 102]], 102, [[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]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, s], [[[a0, t], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a1, s], [[[a1, t], 25], [[s, a0], 24], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, a1, s], [[[a0, t], 24], [[a1, t], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a2, s], [[[a2, t], 26], [[s, a0], 24], [[s, a1], 25], [[s, a3], 27]], 102], [[a0, a2, s], [[[a0, t], 24], [[a2, t], 26], [[s, a1], 25], [[s, a3], 27]], 102], [[a1, a2, s], [[[a1, t], 25], [[a2, t], 26], [[s, a0], 24], [[s, a3], 27]], 102], [[a0, a1, a2, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[s, a3], 27]], 102], [[a3, s], [[[a3, t], 27], [[s, a0], 24], [[s, a1], 25], [[s, a2], 26]], 102], [[a0, a3, s], [[[a0, t], 24], [[a3, t], 27], [[s, a1], 25], [[s, a2], 26]], 102], [[a1, a3, s], [[[a1, t], 25], [[a3, t], 27], [[s, a0], 24], [[s, a2], 26]], 102], [[a0, a1, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a3, t], 27], [[s, a2], 26]], 102], [[a2, a3, s], [[[a2, t], 26], [[a3, t], 27], [[s, a0], 24], [[s, a1], 25]], 102], [[a0, a2, a3, s], [[[a0, t], 24], [[a2, t], 26], [[a3, t], 27], [[s, a1], 25]], 102], [[a1, a2, a3, s], [[[a1, t], 25], [[a2, t], 26], [[a3, t], 27], [[s, a0], 24]], 102], [[a0, a1, a2, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[a3, t], 27]], 102]], 102, [[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]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0073/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 74. The Minimum Cut Is Selected from the Complete Cut Family

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, s], [[[a0, t], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, s], [[[a1, t], 26], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, s], [[[a0, t], 25], [[a1, t], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a2, s], [[[a2, t], 27], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a2, s], [[[a0, t], 25], [[a2, t], 27], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, a2, s], [[[a1, t], 26], [[a2, t], 27], [[s, a0], 25], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, a2, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a3, s], [[[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a3, s], [[[a0, t], 25], [[a3, t], 28], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a1, a3, s], [[[a1, t], 26], [[a3, t], 28], [[s, a0], 25], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a1, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[s, a2], 27], [[s, a4], 29]], 135], [[a2, a3, s], [[[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a4], 29]], 135], [[a0, a2, a3, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[s, a1], 26], [[s, a4], 29]], 135], [[a1, a2, a3, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a4], 29]], 135], [[a0, a1, a2, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a4], 29]], 135], [[a4, s], [[[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a4, s], [[[a0, t], 25], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a1, a4, s], [[[a1, t], 26], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a1, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a4, t], 29], [[s, a2], 27], [[s, a3], 28]], 135], [[a2, a4, s], [[[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28]], 135], [[a0, a2, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a4, t], 29], [[s, a1], 26], [[s, a3], 28]], 135], [[a1, a2, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a3], 28]], 135], [[a0, a1, a2, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a3], 28]], 135], [[a3, a4, s], [[[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27]], 135], [[a0, a3, a4, s], [[[a0, t], 25], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27]], 135], [[a1, a3, a4, s], [[[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27]], 135], [[a0, a1, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a2], 27]], 135], [[a2, a3, a4, s], [[[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26]], 135], [[a0, a2, a3, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26]], 135], [[a1, a2, a3, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25]], 135], [[a0, a1, a2, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29]], 135]], 135, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, s], [[[a0, t], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, s], [[[a1, t], 26], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, s], [[[a0, t], 25], [[a1, t], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a2, s], [[[a2, t], 27], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a2, s], [[[a0, t], 25], [[a2, t], 27], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, a2, s], [[[a1, t], 26], [[a2, t], 27], [[s, a0], 25], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, a2, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a3, s], [[[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a3, s], [[[a0, t], 25], [[a3, t], 28], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a1, a3, s], [[[a1, t], 26], [[a3, t], 28], [[s, a0], 25], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a1, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[s, a2], 27], [[s, a4], 29]], 135], [[a2, a3, s], [[[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a4], 29]], 135], [[a0, a2, a3, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[s, a1], 26], [[s, a4], 29]], 135], [[a1, a2, a3, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a4], 29]], 135], [[a0, a1, a2, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a4], 29]], 135], [[a4, s], [[[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a4, s], [[[a0, t], 25], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a1, a4, s], [[[a1, t], 26], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a1, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a4, t], 29], [[s, a2], 27], [[s, a3], 28]], 135], [[a2, a4, s], [[[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28]], 135], [[a0, a2, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a4, t], 29], [[s, a1], 26], [[s, a3], 28]], 135], [[a1, a2, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a3], 28]], 135], [[a0, a1, a2, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a3], 28]], 135], [[a3, a4, s], [[[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27]], 135], [[a0, a3, a4, s], [[[a0, t], 25], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27]], 135], [[a1, a3, a4, s], [[[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27]], 135], [[a0, a1, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a2], 27]], 135], [[a2, a3, a4, s], [[[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26]], 135], [[a0, a2, a3, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26]], 135], [[a1, a2, a3, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25]], 135], [[a0, a1, a2, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29]], 135]], 135, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0074/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 75. The Minimum Cut Is Selected from the Complete Cut Family

Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[26, 26, 27, 27, 31]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

View answer and explanation([[[s], [[[s, a0], 26], [[s, a1], 27]], 53], [[a0, s], [[[a0, t], 26], [[s, a1], 27]], 53], [[a1, s], [[[a1, t], 27], [[s, a0], 26]], 53], [[a0, a1, s], [[[a0, t], 26], [[a1, t], 27]], 53]], 53, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)

Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 26], [[s, a1], 27]], 53], [[a0, s], [[[a0, t], 26], [[s, a1], 27]], 53], [[a1, s], [[[a1, t], 27], [[s, a0], 26]], 53], [[a0, a1, s], [[[a0, t], 26], [[a1, t], 27]], 53]], 53, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0075/)

Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate





## 76. An Equal-Valued Feasible Flow and Cut Certify Both Optima

Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[2, 2, 3, 3, 7], candidate flows=[2, 2, 3, 3, 0]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(5, 5, [[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)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (5, 5, [[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0076/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 77. An Equal-Valued Feasible Flow and Cut Certify Both Optima

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[3, 3, 4, 4, 5, 5], candidate flows=[3, 3, 4, 4, 5, 5]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(12, 12, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (12, 12, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0077/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 78. An Equal-Valued Feasible Flow and Cut Certify Both Optima

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=[4, 4, 5, 5, 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(22, 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]], [[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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (22, 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]], [[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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0078/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 79. An Equal-Valued Feasible Flow and Cut Certify Both Optima

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10], candidate flows=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 0]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(35, 35, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (35, 35, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0079/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 80. An Equal-Valued Feasible Flow and Cut Certify Both Optima

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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(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)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0080/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 81. An Equal-Valued Feasible Flow and Cut Certify Both Optima

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[7, 7, 8, 8, 9, 9], candidate flows=[7, 7, 8, 8, 9, 9]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(24, 24, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (24, 24, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0081/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 82. An Equal-Valued Feasible Flow and Cut Certify Both Optima

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=[8, 8, 9, 9, 10, 10, 11, 11, 13], candidate flows=[8, 8, 9, 9, 10, 10, 11, 11, 0]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(38, 38, [[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]], [[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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (38, 38, [[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]], [[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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0082/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 83. An Equal-Valued Feasible Flow and Cut Certify Both Optima

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13], candidate flows=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(55, 55, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (55, 55, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0083/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 84. An Equal-Valued Feasible Flow and Cut Certify Both Optima

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=[10, 10, 11, 11], candidate flows=[10, 10, 11, 11]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(21, 21, [[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)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (21, 21, [[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0084/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 85. An Equal-Valued Feasible Flow and Cut Certify Both Optima

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=[11, 11, 12, 12, 13, 13, 16], candidate flows=[11, 11, 12, 12, 13, 13, 0]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(36, 36, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (36, 36, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0085/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 86. An Equal-Valued Feasible Flow and Cut Certify Both Optima

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], candidate flows=[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 (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(54, 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]], [[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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (54, 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]], [[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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0086/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 87. An Equal-Valued Feasible Flow and Cut Certify Both Optima

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17], candidate flows=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(75, 75, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (75, 75, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0087/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 88. An Equal-Valued Feasible Flow and Cut Certify Both Optima

Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[14, 14, 15, 15, 19], candidate flows=[14, 14, 15, 15, 0]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(29, 29, [[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)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (29, 29, [[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0088/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 89. An Equal-Valued Feasible Flow and Cut Certify Both Optima

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[15, 15, 16, 16, 17, 17], candidate flows=[15, 15, 16, 16, 17, 17]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(48, 48, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (48, 48, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0089/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 90. An Equal-Valued Feasible Flow and Cut Certify Both Optima

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=[16, 16, 17, 17, 18, 18, 19, 19], candidate flows=[16, 16, 17, 17, 18, 18, 19, 19]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(70, 70, [[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]], [[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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (70, 70, [[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]], [[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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0090/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 91. An Equal-Valued Feasible Flow and Cut Certify Both Optima

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22], candidate flows=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 0]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(95, 95, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (95, 95, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0091/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 92. An Equal-Valued Feasible Flow and Cut Certify Both Optima

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=[18, 18, 19, 19], candidate flows=[18, 18, 19, 19]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(37, 37, [[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)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (37, 37, [[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0092/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 93. An Equal-Valued Feasible Flow and Cut Certify Both Optima

Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[19, 19, 20, 20, 21, 21], candidate flows=[19, 19, 20, 20, 21, 21]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(60, 60, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (60, 60, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0093/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 94. An Equal-Valued Feasible Flow and Cut Certify Both Optima

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=[20, 20, 21, 21, 22, 22, 23, 23, 0]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(86, 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]], [[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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (86, 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]], [[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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0094/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 95. An Equal-Valued Feasible Flow and Cut Certify Both Optima

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25], candidate flows=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(115, 115, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (115, 115, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0095/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 96. An Equal-Valued Feasible Flow and Cut Certify Both Optima

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=[22, 22, 23, 23], candidate flows=[22, 22, 23, 23]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(45, 45, [[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)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (45, 45, [[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0096/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 97. An Equal-Valued Feasible Flow and Cut Certify Both Optima

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], candidate flows=[23, 23, 24, 24, 25, 25, 0]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(72, 72, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (72, 72, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0097/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 98. An Equal-Valued Feasible Flow and Cut Certify Both Optima

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=[24, 24, 25, 25, 26, 26, 27, 27], candidate flows=[24, 24, 25, 25, 26, 26, 27, 27]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(102, 102, [[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]], [[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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (102, 102, [[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]], [[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]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0098/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 99. An Equal-Valued Feasible Flow and Cut Certify Both Optima

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29], candidate flows=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(135, 135, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (135, 135, [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[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], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0099/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate





## 100. An Equal-Valued Feasible Flow and Cut Certify Both Optima

Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[26, 26, 27, 27, 31], candidate flows=[26, 26, 27, 27, 0]. 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).

Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.

View answer and explanation(53, 53, [[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)

First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (53, 53, [[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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0100/)

Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations/validation-report.json)