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In [Ni] we showed that if a commuting square C satisfies a certain span condition then it is isolated among all commuting squares, modulo isomorphisms. We now ask the converse question: if the span condition fails, is C a limit of non-isomorphic commuting squares? We find new necessary conditions for C to be such a limit point. We give an application to the classification of complex Hadamard matrices. Our result can be used to argue that certain 1-parameter families of Hadamard matrices can notdoi:10.1512/iumj.2011.60.4294 fatcat:up3irafdy5ftrlxfmzckrdn6hm