  We show that Zamolodchikov dynamics of a recurrent quiver has zero algebraic
entropy only if the quiver has a weakly subadditive labeling, and conjecture
the converse. By assigning a pair of generalized Cartan matrices of affine type
to each quiver with an additive labeling, we completely classify such quivers,
obtaining $40$ infinite families and $13$ exceptional quivers. This completes
the program of classifying Zamolodchikov periodic and integrable quivers.
