Suppose that $\phi(x)$ is a $\Delta^{1}_{2}$-formula (without parameters) and let $A:=\{x\subseteq\omega:\phi(x)\}$. It is clear that, e.g. if there are Cohen-generics over $L$, then $A$ cannot be the set of constructible reals, roughly because $\Delta^{1}_{2}$ cannot distinguish between sufficiently high Cohen-generics in $L$ from Cohen-generics over $L$. I wonder whether this can be strengthenend in the following way:

Given appropriate largeness assumptions (existence of generic filters, large cardinals...), at least one of $A$ and $\mathfrak{P}(\omega)\setminus A$ contains real numbers of all degrees of constructibility. In other words, can $\Delta^{1}_{2}$ "separate" degrees of constructibility in a nontrivial way?

Suppose that $\phi(x)$ is a $\Delta^{1}_{2}$-formula (without parameters) and let $A:=\{x\subseteq\omega:\phi(x)\}$. It is clear that, e.g. if there are Cohen-generics over $L$, then $A$ cannot be the set of constructible reals, roughly because $\Delta^{1}_{2}$ cannot distinguish between sufficiently high Cohen-generics in $L$ and Cohen-generics over $L$. I wonder whether this can be strengthenend in the following way:

Given appropriate largeness assumptions (existence of generic filters, large cardinals...), at least one of $A$ and $\mathfrak{P}(\omega)\setminus A$ contains real numbers of all degrees of constructibility. In other words, can $\Delta^{1}_{2}$ "separate" degrees of constructibility in a nontrivial way?