  Richard Eager and Sebastian Franco introduced a change of basis
transformation on the F-polynomials of Fomin and Zelevinsky, corresponding to
rewriting them in the basis given by fractional brane charges rather than
quiver gauge groups. This transformation seems to display a surprising
stabilization property, apparently causing the first few terms of the
polynomials at each step of the mutation sequence to coincide. Eager and Franco
conjecture that this transformation will always cause the polynomials to
converge to a formal power series as the number of mutations goes to infinity,
at least for quivers possessing certain symmetries and along periodic mutation
sequences respecting such symmetries. In this paper, we verify this convergence
in the case of the Kronecker and Conifold quivers. We also investigate
convergence in the $F_0$ quiver. We provide a combinatorial interpretation for
the stable cluster variables in each appropriate case.
