  McCarthy's Theorem for the mapping class group of a closed hyperbolic surface
states that for any two mapping classes $\sigma,\tau \in \mathrm{Mod}(S)$ there
is some power $N$ such that the group $\langle \sigma^N,\tau^N\rangle$ is
either free of rank two or abelian, and gives a geometric criterion for the
dichotomy. The analogous statement is false in linear groups, and unresolved
for outer automorphisms of a free group. Several analogs are known for
exponentially growing outer automorphisms satisfying various technical
hypothesis. In this article we prove an analogous statement when $\sigma$ and
$\tau$ are linearly growing outer automorphisms of $F_r$, and give a geometric
criterion for the dichotomy. Further, Hamidi-Tehrani proved that for Dehn
twists in the mapping class group this independence dichotomy is
\emph{uniform}: $N=4$ suffices. In a similar style, we obtain an $N$ that
depends only on the rank of the free group.
