Balanced Laurent Potentials for Finite Cluster Actions

We give counterexamples to the uniqueness assertion for maximum potentials in Theorem 4.7 of Ishibashi's Fixed point theorem for cluster modular groups, arXiv:2603.14338v2. Both a rank-one example and a full-rank type A2 example satisfy its filling conditions and have nontrivial finite modular groups, but their maximum potentials have continua of minimizers. More generally, adjoining a sufficiently large positive integral constant can make any prescribed compact set consist of minimizers. The finite-subgroup fixed-point conclusion is nevertheless preserved: the group-averaged sum of positive Laurent functions with balanced support is coercive and strictly convex in a global logarithmic seed chart. Its unique minimizer is fixed by the group. The logarithm of this sum has the same properties. This gives a replacement proof of the credited fixed-point conclusion using balanced support alone, without the additional slope-span condition. No properness result for infinite cluster modular groups or closure of the original AIM braid-variety question is claimed. This English preprint is AI-assisted, self-audited and unrefereed. Novelty remains undetermined; no independent human review, formal verification or absolute-priority claim is made.