Incremental universality of Wigner random matrices

Properties of universality have essential relevance for the theory of random matrices usually called the Wigner ensemble. The issue was analysed up to recent years with detailed and relevant results. We present a slightly different view and compare the large-$n$ limit of connected correlators of distinct ensembles: universality has steps or degrees, precisely counted by the number of probability moments of the matrix entries, which match among distinct ensembles.