  This article continues the investigation of the tracial geometry of
classifiable $\mathrm{C}^*$-algebras that have real rank zero and stable rank
one. Using the language of optimal transport, we describe several situations in
which the distance between unitary orbits of $^*$-homomorphisms into such
algebras can be computed in terms of tracial data. The domains we consider are
certain (noncommutative) CW complexes, and the measurement is relative to a
family of self-adjoint elements that are in a suitable sense tracially
Lipschitz. As another application of the utility of this Lipschitz structure,
we show how such elements can be repurposed to witness statistical features of
endomorphisms in the classifiable category, in particular the tracial version
of the (almost-sure) central limit theorem.
