  This is the fourth in a series of articles devoted to showing that a typical
covering map of large degree to a fixed, regular graph has its new adjacency
eigenvalues within the bound conjectured by Alon for random regular graphs.
  In this paper we prove a {\em Sidestepping Theorem} that is more general and
easier to use than earlier theorems of this kind. Such theorems concerns a
family probability spaces $\{{\mathcal{M}}_n\}$ of $n\times n$ matrices, where
$n$ varies over some infinite set, $N$, of natural numbers. Many trace methods
use simple "Markov bounds" to bound the expected spectral radius of elements of
${\mathcal{M}}_n$: this consists of choosing one value, $k=k(n)$, for each
$n\in N$, and proving expected spectral radius bounds based on the expected
value of the trace of the $k=k(n)$-power of elements of ${\mathcal{M}}_n$. {\em
Sidestepping} refers to bypassing such simple Markov bounds, obtaining improved
results using a number of values of $k$ for each fixed $n\in N$.
  In more detail, if the $M\in {\mathcal{M}}_n$ expected value of ${\rm
Trace}(M^k)$ has an asymptotic expansion in powers of $1/n$, whose coefficients
are "well behaved" functions of $k$, then one can get improved bounds on the
spectral radius of elements of ${\mathcal{M}}_n$ that hold with high
probability. Such asymptotic expansions are shown to exist in the third article
in this series for the families of matrices that interest us; in the fifth and
sixth article in this series we will apply the Sidestepping Theorem in this
article to prove the main results in this series of articles.
  This article is independent of all other articles in this series; it can be
viewed as a theorem purely in probability theory, concerning random matrices
or, equivalently, the $n$ random variables that are the eigenvalues of the
elements of ${\mathcal{M}}_n$.
