  We present a renormalizable theory of scalars in which the low energy
effective theory contains a pseudo-Goldstone Boson with a compact field space
of 2{\pi} F and an approximate discrete shift symmetry Z_Q with Q>>1, yet the
number of fields in the theory goes as log(Q). Such a model can serve as a UV
completion to models of relaxions and is a new source of exponential scale
separation in field theory. While the model is local in `theory space', it
appears to not have a continuum generalization (i.e., it cannot be a
deconstructed extra dimension). Our framework shows that super-Planckian field
excursions can be mimicked while sticking to renormalizable four-dimensional
quantum field theory. We show that a supersymmetric extension is
straightforwardly obtained and we illustrate possible UV completions based on a
compact extra-dimension, where all global symmetries arise accidentally as a
consequence of gauge invariance and 5D locality.
