 Simplicial section compression

Split a set family in the hypercube graph according to whether coordinate $i$ is present, and delete that coordinate from the present section. Replace both sections by initial segments of the simplicial order on the discrete cube with the same respective sizes. Induction and nesting of initial-segment closed graph neighbourhoods show that this operation cannot enlarge the original closed graph neighbourhood. Repeated nontrivial compressions terminate because the sum of simplicial positions strictly decreases.