A Torus with Eleven Convex Quadrilateral Faces

Grünbaum and Szilassi conjectured in 2009 that a toroidal polyhedron cannot have ten or eleven convex quadrilateral faces. We give an explicit counterexample with eleven faces and eleven integer vertices. Every face is planar and strictly convex, adjacent faces lie in different planes, and distinct faces meet in their prescribed common edge or vertex, or are disjoint. The construction has a simple geometric interpretation: remove two triangular caps from a convex polyhedron and join their boundaries with the three walls of an interior triangular prism. We describe the direct search route, the passage from numerical coordinates to exact incidences, and the geometric certificate. A small standalone package verifies the witness using rational arithmetic and no third-party libraries.