  We show the invalidity of finitary counterparts for three classification
theorems: The preservation of being a Bernoulli shift through factors, Sinai's
factor theorem, and the weak Pinsker property. We construct a finitary factor
of an i.i.d. process which is not finitarily isomorphic to an i.i.d. process,
showing that being finitarily Bernoulli is not preserved through finitary
factors. This refutes a conjecture of M. Smorodinsky [11], which was first
suggested by D. Rudolph [7]. We further show that any ergodic system is
isomorphic to a process none of whose finitary factors are i.i.d. processes,
and in particular, there is no general finitary Sinai's factor theorem for
ergodic processes. An immediate consequence of this result is the invalidity of
a finitary weak Pinsker property, answering a question of G. Pete and T. Austin
[1].
