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.

References

To obtain a full additive categorification of the cluster algebras arising from higher Teichmüller theory, the following two tasks remain to be completed: * Show that the ice quiver potential described in \Cref{def:surface_ice_quiver} is non-degenerate.

— Cluster theory of topological Fukaya categories. Part II: Higher Teichmüller theory  (2510.05925 - Christ, 7 Oct 2025) in Introduction, Subsection 1.4 (The Higgs category and cluster tilting theory)

Li and Peng obtained Jacobi-finite potentials on a new class of $2$-acyclic quivers by a covering construction, while leaving the corresponding non-degeneracy assertion as a conjecture.

— Obstructions to Jacobi-Finiteness of Quivers with Potentials  (2609.11200 - Chang et al., 10 Sep 2026) in Section 1, Introduction