  Exact diagonalization studies have revealed that the energy spectrum of
interacting electrons in the lowest Landau level splits, non-perturbatively,
into bands, which is responsible for the fascinating phenomenology of this
system. The theory of nearly free composite fermions has been shown to be valid
for the lowest band, and thus to capture the low temperature physics, but it
over-predicts the number of states for the excited bands. We explain the state
counting of higher bands in terms of composite fermions with an infinitely
strong short range interaction between an excited composite-fermion particle
and the hole it leaves behind. This interaction, the form of which we derive
from the microscopic composite fermion theory, eliminates configurations
containing certain tightly bound composite-fermion excitons. With this
modification, the composite-fermion theory reproduces,for all well-defined
excited bands seen in exact diagonalization studies, an exact counting for
$\nu>1/3$, and an almost exact counting for $\nu\leq 1/3$. The resulting
insight clarifies that the corrections to the nearly free composite fermion
theory are not thermodynamically significant at sufficiently low temperatures,
thus providing a microscopic explanation for why it has proved successful for
the analysis of the various properties of the composite-fermion Fermi sea.
