We prove global smooth classification results for Anosov ℤk actions on general compact manifolds, under certain irreduciblity conditions and the presence of sufficiently many Anosov elements. In particular, we remove all the uniform control assumptions which were used in all the previous results towards the Katok-Spatzier global rigidity conjecture on general manifolds. The main idea is to create a new mechanism labelled nonuniform redefining argument, to prove continuity of certain dynamically-defined objects. This leads to uniform control for higher-rank actions and should apply to more general rigidity problems in dynamical systems.