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We show that any II$_1$ factor that has the same 4-quantifier theory as thehyperfinite II$_1$ factor $\mathcal{R}$ satisfies the conclusion of the PopaFactorial Commutant Embedding Problem (FCEP) and has the Brown property. Theseresults improve recent results proving the same conclusions under the strongerassumption that the factor is actually elementarily equivalent to$\mathcal{R}$. In the same spirit, we improve a recent result of thefirst-named author, who showed that if (1) the amalgamated free product ofembeddable factors over a property (T) base is once again embeddable, and (2)$\mathcal{R}$ is an infinitely generic embeddable factor, then the FCEP is trueof all property (T) factors. In this paper, it is shown that item (2) can beweakened to assume that $\mathcal{R}$ has the same 3-quantifier theory as aninfinitely generic embeddable factor.

Properties expressible in small fragments of the theory of the hyperfinite II_1 factor