  For any tracial non-commutative probability space $(\mathcal{A}, \varphi)$,
C\'{e}bron, Dahlqvist, and Male showed that one can always construct an
enveloping traffic space $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ that
extends the trace. This construction provides a universal object that allows
one to appeal to the traffic probability framework in generic situations,
prioritizing an understanding of its structure. In this article, we prove that
$(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ admits a canonical free product
decomposition $\mathcal{A} * \mathcal{A}^\intercal *
\Theta(\mathcal{G}(\mathcal{A}))$. In particular, $\mathcal{A}^\intercal$ is an
anti-isomorphic copy of $\mathcal{A}$, and $\Theta(\mathcal{G}(\mathcal{A}))$
is, up to degeneracy, a commutative algebra generated by Gaussian random
variables with a covariance structure diagonalized by the graph operations. If
$(\mathcal{A}, \varphi)$ itself is a free product, then we describe how this
additional structure lifts into $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$.
Here, we find a connection between free independence and classical independence
opposite the usual direction. Up to degeneracy, we further show that
$(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ is spanned by tree-like graph
operations. Finally, we apply our results to the study of large (possibly
dependent) random matrices. Our analysis relies on the combinatorics of cactus
graphs and the resulting cactus-cumulant correspondence.
