Growth rates of permutation classes: Categorization up to the uncountability threshold

In the antecedent paper to this it was established that there is an algebraic number ξ ≈ 2.30522 such that while there are uncountably many growth rates of permutation classes arbitrarily close to ξ, there are only countably many less than ξ. Here we provide a complete characterization of the growth rates less than ξ. In particular, this classification establishes that ξ is the least accumulation point from above of growth rates and that all growth rates less than or equal to ξ are achieved by finitely based classes. A significant part of this classification is achieved via a reconstruction result for sum indecomposable permutations. We conclude by refuting a suggestion of Klazar, showing that ξ is an accumulation point from above of growth rates of finitely based permutation classes.