Isometric rigidity of complete Pick algebras on monomial varieties

We establish a geometric rigidity theorem for a class of monomial varieties in finite-dimensional unit balls. Every ball automorphism mapping one such variety onto another is unitary, and whenever such an equivalence exists, it can also be realized by a permutation of the ambient coordinates. As a consequence, we obtain a complete isometric classification of the multiplier algebras of integrally generated complete Nevanlinna–Pick Dirichlet series kernels in terms of their defining weight and frequency data. We also obtain an explicit isometric classification for a class of complete Nevanlinna–Pick kernels on bounded Reinhardt domains.