Complexity of $Σ_n$ theory of $H(ω_2)$ under MM$^{++}$
Under Woodin's $\mathbb{P}_\text{max}$ axiom (which is implied by MM$^{++}$), what is the complexity of the $Σ_n$ theory of $H(ω_2)$? Same question for $Σ_n(I_\text{NS})$ (i.e. using the nonstationary ideal on $ω_1$).
Update (Nov 24, 2024): I posed the question to Hugh Woodin, and he replied that, as he recalls it, the exact complexity, even with real parameters, of $Σ_n$ theory of $H(ω_2)$ here for $n≥2$ is $Σ^1_{n+2}$, and $Σ_n(I_\text{NS})$ theory has the same complexity.
Surprisingly, under large cardinal axioms, there is a generic extension of $V$ such that the theories of $H(ω_1)$ and $H(ω_2)$ have the same Turing degree. Specifically, assuming a supercompact cardinal, some generic extension of $V$ satisfies Martin's Maximum$^{++}$ (MM$^{++}$). In 2019, MM$^{++}$ was shown to imply Woodin's $\mathbb{P}_\text{max}$ axiom (which Woodin simply called "(*)") (Martin’s Maximum++ implies Woodin’s Axiom (*) by David Aspero and and Ralf Schindler). In turn, the $\mathbb{P}_\text{max}$ axiom (which is consistent relative to $ω$ Woodin cardinals, but was not initially known to hold in a generic extension of $V$) implies that the theories of $H(ω_1)$ and $H(ω_2)$ have the same Turing degree.
I am convinced that CH is true (as opposed to $c=ω_2$ under PFA), but it is interesting to explore other 'universes', and ask, under the axiom of choice, what happens if we try to minimize the number of 'pathologies' of size $ω_1$. PFA and its extensions MM and MM$^{++}$ give a plausible answer. MM$^{++}$ asserts that for every forcing $P$ preserving stationarity of subsets of $ω_1$, for every collection of $ω_1$ dense subsets of $P$, and for every collection of $ω_1$ names for stationary subsets of $ω_1$, there is a filter that meets all the dense subsets and realizes all the names by actual stationary sets.
Under ZF + AD, every subset of $ω_1$ is constructible from a real (and every real has a sharp), so $Σ_n^{H(ω_2)}$ (allowing $ω_1$ as a constant for $n=1$) and $Σ_n^{H(ω_2)}(I_\text{NS})$ have the same complexity as $Σ^1_{n+2}$, even allowing real parameters. I do not know what happens under the $\mathbb{P}_\text{max}$ axiom.
A generalization: Assuming a proper class of Woodin cardinals, a natural generalization of $\mathbb{P}_\text{max}$ axiom (which I think follows from MM$^{++}$ under a proper class of Woodin cardinals) is: For every universally Baire $A$, every $Ω$-consistent $Π_2$ sentence in $(H(ω_2),∈,I_\text{NS},A)$ holds. For this generalization and a uB $A$, we can ask about the complexity of $Σ_n(A)$ and $Σ_n(A,I_\text{NS})$ theories of $H(ω_2)$, including whether the theories are $Σ^1_{n+2}(A)$.
