  Let $E$ be an analytic equivalence relation on a Polish space. We introduce a
framework for studying the possible "reasonable" complete classifications and
the complexity of possible classifying invariants for $E$, such that: (1) the
standard results and intuitions regarding classifications by countable
structures are preserved in this framework; (2) this framework respects Borel
reducibility; (3) this framework allows for a precise study of the possible
invariants of certain equivalence relations which are not classifiable by
countable structures, such as $E_1$.
  In this framework we show that $E_1$ can be classified, with classifying
invariants which are $\kappa$-sequences of $E_0$-classes where
$\kappa=\mathfrak{b}$, and it cannot be classified in such a manner if
$\kappa<\mathbf{add}(\mathcal{B})$.
  These results depend on analyzing the following sub-model of a Cohen real
extension, introduced by Kanovei-Sabok-Zapletal (2013) and Larson-Zapletal
(2020). Let $\left<c_n:\,n<\omega\right>$ be a generic sequence of Cohen reals,
and define the tail intersection model
  $$M=\bigcap_{n<\omega}V[\left<c_m:\,m\geq n\right>].$$ An analysis of reals
in $M$ will provide lower bounds for the possible invariants for $E_1$.
  We also extend the characterization of turbulence from Larson-Zapletal (2020)
in terms of intersection models.
