Limit points of commuting squares Remus Nicoara 46L3715B34subfactorcommuting squareHadamard matrixmoduli space In an attempt to understand the structure of the moduli space of commuting squares, we ask the question: When is a commuting square $\mathfrak{C}$ a limit of non-isomorphic commuting squares? We present necessary second-order conditions on such a $\mathfrak{C}$. 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. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4294 10.1512/iumj.2011.60.4294 en Indiana Univ. Math. J. 60 (2011) 847 - 858 state-of-the-art mathematics http://iumj.org/access/