Positive polytopes and the full Fock–Goncharov conjecture

We prove the origin-interior form of the polytope criterion of Gross, Hacking, Keel and Kontsevich: over a field of characteristic zero, full Fock–Goncharov for a principal coefficient cluster variety is equivalent to the existence of a bounded full-dimensional positive polytope containing the origin in its interior. Replacing the interior condition by membership of the origin gives a false criterion.