 Weighted cut lemma for commutative graphs

A minimum-weight set meeting every bottom-to-top path can be chosen entirely in the endpoint layers. The local step is a three-layer channel whose middle layer is a minimum cut. Replacing middle subsets by their successors or predecessors gives expansion inequalities. Sorting middle vertices by incoming degree and top vertices by their largest predecessor degree gives one edge-count inequality; reversing the graph gives the opposite inequality. Equality forces the bottom layer to have the same weight as the middle cut. Replacing the highest intermediate portion of a minimum cut by this bottom layer repeatedly proves the assertion.