  We study tropical friezes and cluster-additive functions associated to
symmetrizable generalized Cartan matrices in the framework of Fock-Goncharov
duality in cluster algebras. In particular, we generalize and prove a
conjecture of C. M. Ringel on cluster-additive functions associated to
arbitrary Cartan matrices of finite type. For a Fock-Goncharov dual pair of
positive spaces of finite type, we use tropical friezes and cluster-additive
functions to explicitly express the Fock-Goncharov pairing between their
tropical points and the bijections between global monomials on one and tropical
points of the other.
