Covering entropy for types in tracial $\mathrm{W}^*$-algebras

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 $h(\mathcal{M}) \leq 0$.