  We show that any finite-entropy, countable-valued finitary factor of an i.i.d
process can also be expressed as a finitary factor of a finite-valued i.i.d
process whose entropy is arbitrarily close to the target process. As an
application, we give an affirmative answer to a question of van den Berg and
Steif about the critical Ising model on $\mathbb{Z}^d$. En route, we prove
several results about finitary isomorphisms and finitary factors. Our results
are developed in a new framework for processes invariant to a permutation group
of a countable set satisfying specific properties. This new framework includes
all ``classical'' processes over countable amenable groups and all invariant
processes on transitive amenable graphs with ``uniquely centered balls''. Some
of our results are new already for $\mathbb{Z}$-processes. We prove a relative
version of Smorodinsky's isomorphism theorem for finitely dependent
$\mathbb{Z}$-processes. We also extend the Keane--Smorodinsky finitary
isomorphism theorem to countable-valued i.i.d processes and to i.i.d processes
taking values in a Polish space.
