Properties expressible in small fragments of the theory of the hyperfinite II$_1$ factor

We show that any II ${}_1$ factor that has the same 4-quantifier theory as the hyperfinite II ${}_1$ factor $ℛ$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These results improve recent results proving the same conclusions under the stronger assumption that the factor is actually elementarily equivalent to $ℛ$ . In the same spirit, we improve a recent result of the first-named author, who showed that if (1) the amalgamated free product of embeddable factors over a property (T) base is once again embeddable, and (2) $ℛ$ is an infinitely generic embeddable factor, then the FCEP is true of all property (T) factors. In this paper, it is shown that item (2) can be weakened to assume that $ℛ$ has the same 3-quantifier theory as an infinitely generic embeddable factor.