  We study embeddings of tracial $\mathrm{W}^*$-algebras into a ultraproduct of
matrix algebras through an amalgamation of free probabilistic and
model-theoretic techniques. Jung implicitly and Hayes explicitly defined
$1$-bounded entropy through the asymptotic covering numbers of Voiculescu's
microstate spaces, that is, spaces of matrix tuples
$(X_1^{(N)},X_2^{(N)},\dots)$ having approximately the same $*$-moments as the
generators $(X_1,X_2,\dots)$ of a given tracial $\mathrm{W}^*$-algebra. We
study the analogous covering entropy for microstate spaces defined through
formulas that use not only $*$-algebra operations and the trace, but also
suprema and infima, such as arise in the model theory of tracial
$\mathrm{W}^*$-algebras initiated by Farah, Hart, and Sherman. By relating the
new theory with the original $1$-bounded entropy, we show that if
$h(\mathcal{N}:\mathcal{M}) \geq 0$, then there exists an embedding of
$\mathcal{M}$ into a matrix ultraproduct $\mathcal{Q} = \prod_{n \to
\mathcal{U}} M_n(\mathbb{C})$ such that $h(\mathcal{N}:\mathcal{Q})$ is
arbitrarily close to $h(\mathcal{N}:\mathcal{M})$. We deduce if all embeddings
of $\mathcal{M}$ into $\mathcal{Q}$ are automorphically equivalent, then
$\mathcal{M}$ is strongly $1$-bounded and in fact has $h(\mathcal{M}) \leq 0$.
