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This paper shows the polytopality of any finite type $\mathbf{g}$-vector fan,acyclic or not. In fact, for any finite Dynkin type $\Gamma$, we construct auniversal associahedron $\mathsf{Asso}_{\mathrm{un}}(\Gamma)$ with the propertythat any $\mathbf{g}$-vector fan of type $\Gamma$ is the normal fan of asuitable projection of $\mathsf{Asso}_{\mathrm{un}}(\Gamma)$.

Polytopal realizations of finite type $\mathbf{g}$-vector fans