  We prove sharp spectral transition in the arithmetics of phase between
localization and singular continuous spectrum for Diophantine almost Mathieu
operators. We also determine exact exponential asymptotics of eigenfunctions
and of corresponding transfer matrices throughout the localization region. This
uncovers a universal structure in their behavior governed by the exponential
phase resonances. The structure features a new type of hierarchy, where
self-similarity holds upon alternating reflections.
