Non-degeneracy of the amalgamation ice quiver potential

Show that the ice quiver potential defined for the amalgamation ice quiver in Definition 5.?? (Cref{def:surface_ice_quiver}) is non-degenerate, meaning that under iterated mutations of the ice quiver with potential at non-frozen vertices no 2-cycles appear.

Background

The paper introduces an amalgamation ice quiver with potential (Definition Cref{def:surface_ice_quiver}) obtained by gluing triangle ice quivers along a triangulation. For categorical cluster characters to be well-behaved, the potential must be non-degenerate, i.e., iterated mutations at non-frozen vertices should not produce 2-cycles or loops. Earlier in the introduction, the authors note that non-degeneracy ensures a bijection between reachable rigid objects and cluster variables via cluster characters.

Establishing non-degeneracy of the constructed potential would validate the mutation-compatibility and combinatorial properties required to complete the additive categorification of the cluster algebras associated with higher Teichmüller spaces.