  We establish that there is an algebraic number $\xi\approx 2.30522$ such that
while there are uncountably many growth rates of permutation classes
arbitrarily close to $\xi,$ there are only countably many less than $\xi$.
Central to the proof are various structural notions regarding generalized grid
classes and a new property of permutation classes called concentration. The
classification of growth rates up to $\xi$ is completed in a subsequent paper.
