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We complete the investigation of the Gibbs properties of the fuzzy Pottsmodel on the d-dimensional torus with Kac interaction which was started byJahnel and one of the authors. As our main result of the present paper, weextend the previous sharpness result of mean-field bounds to cover all possiblecases of fuzzy transformations, allowing also for the occurrence of Isingclasses. The closing of this previously left open Ising-gap involves ananalytical argument showing uniqueness of minimizing profiles for certainnon-homogeneous conditional variational problems.

Gibbs-non-Gibbs transition in the fuzzy Potts models with a Kac-type interaction: Closing the Ising gap