---

Source. Ford–Fulkerson (1958), Section 1, printed pp. 78–79 and references 1–5 on p. 84 (published scan).

The representative proofs use the following external theorem. In a finite directed network with nonnegative integer capacities, distinct source $s$ and sink $t$, nonnegative flows satisfying capacity bounds and conservation at the other vertices have a maximum value. That value equals the minimum capacity of a cut $L,V\setminus L$ with $s\in L$, $t\notin L$. Some maximum flow is integral. Flow value is net outflow at $s$; cut capacity counts arcs directed from $L$ to its complement, not arcs crossing in both directions.

The source states these on p. 79 as two cited theorems. Its Minimal cut theorem, for the capacities of p. 78 (each a nonnegative real or plus infinity), cites references 3, 4 and 5 (Dantzig–Fulkerson and Ford–Fulkerson's 1957 and 1956 papers); its Integrity theorem, for integral capacities, cites references 3 and 4. Reference 1 (Dantzig–Orden–Wolfe) is cited only for the simplex method. The paper does not prove either theorem. The directly cited originals include Ford–Fulkerson, A simple algorithm for finding maximal network flows and an application to the Hitchcock problem, Canadian Journal of Mathematics 9 (1957), 210–218, and Maximal flow through a network, Canadian Journal of Mathematics 8 (1956), 399–404. Their proofs remain external to the 1958 article. The exact integer theorem for the terminal directions used by its layered networks is reconstructed in the 1957 residual algorithm. The distinct 1956 saturated-edge proof is reconstructed for finite undirected simple-chain flows with positive real capacities. It does not replace the directed integer theorem needed by these representative networks. The remarks about integral extreme points and simplex algorithms are also pointers, not additional reconstructed results.

The paper allows infinite capacities in its representative networks. Every such arc below is replaced by a finite integer $K>F$, where $F$ is the target flow and also the total capacity leaving the source. This replacement is exact for the tests used here. Every directed path runs forward through a finite list of layers, so a feasible flow decomposes into source–sink paths: repeatedly follow a positive arc, use conservation to continue, and subtract the smallest flow on the resulting path. No cycle is possible; each subtraction makes an arc zero, so the process terminates. Its total path weight is the flow value, at most $F$, and consequently no arc carries more than $F$. Thus $K$ imposes no new restriction. A cut containing one of these arcs has capacity at least $K>F$ and already satisfies the required lower bound; all remaining cuts have the same finite capacity as in the paper.

Only this elementary finite-capacity translation is supplied locally. It does not prove max-flow/min-cut or integrality. The network constructions establish mathematical equivalences and would accept an integral max-flow algorithm as a subroutine; no implementation or complexity claim is verified here.

The introduction's stronger suggestion that all the problems can be reduced to Hall's theorem is expressly left undemonstrated in the paper. It is not counted as an additional proof. The Hoffman–Kuhn and Mann–Ryser citations record earlier representative results; their full papers are not reproduced by the substitutions below.