Phase faces and acyclic sets in the eight-dimensional cube

This research note studies Boolean functions on the eight-dimensional cube for which each vertex has exactly four neighbours of the opposite value. At each vertex, record the four coordinates whose individual changes alter the function's value. The resulting sets define a directed graph. I prove that deleting any set of at most four vertices leaves six vertices whose induced directed graph is acyclic. The proof combines restrictions on four-dimensional faces with an incidence count: each vertex lies in at most forty of the indexed support faces, so a set meeting every such face has at least thirteen vertices. The acyclic six avoids the deleted vertices and therefore survives arbitrary changes to the assigned coordinate sets at those vertices. All proofs are included. The note concerns this specified class of Boolean functions, not arbitrary assignments of coordinate sets. The constants forty and thirteen are not claimed to be optimal. The known Boolean-function framework and classical counting ingredients are credited in the references.