We prove that the Poisson/Gaudin–Mehta phase transition conjectured to occur when the bandwidth of an N×N symmetric band matrix grows like b=N−−√ is naturally observable in the rate of convergence of the level density to the Wigner semi-circle law. Specifically, we show for periodic and non-periodic band matrices the rate of convergence of the fourth moment of the level density is independent of the boundary conditions in the localized regime b≪N−−√ with a rate of O(1b) for both cases, whereas in the delocalized regime b≫N−−√ where boundary effects become important, the rate of convergence for the two ensembles differs significantly, slowing to O(bN) for non-periodic band matrices. Additionally, we examine the case of thick non-periodic band matrices b=cN, showing that the fourth moment is maximally deviated from the Wigner semi-circle law when b=25N, and provide numerical evidence that the eigenvector statistics also exhibit critical behavior at this point.