Every Two-Vertex Premaniplex Is the Symmetry Type Graph of a Finite Abstract Polytope

Abstract

For every rank r>=3, there are 2^r-1 connected two-vertex premaniplexes. We prove that every one of them is the full symmetry type graph of a finite abstract polytope. The main argument localizes the support of the recursive regular-polytopal voltage construction, treats arbitrary sets of internal link colors simultaneously, and establishes the remaining path-intersection identity. Two exceptional rank-four base types are handled by exact finite computation. This upgrades the known finite-maniplex realization theorem to a finite-polytopal realization theorem in every rank at least three. Status: Public Beta; internally verified candidate proof; external mathematical review pending.