Limit points of commuting squares

In an attempt to understand the structure of the moduli space of commuting squares, we ask the question: when is a commuting square C a limit of non-isomorphic commuting squares? We present necessary second order conditions on such a C. We give an application to the classification of complex Hadamard matrices. Such matrices correspond to spin model commuting squares. We exemplify on Petrescu’s matrices how our result can be used to decide if a one-parameter family can be extended to a multi-parametric family of Hadamard matrices.