Are there trees for $(\Sigma^2_1)^{\text{uB}}$?

Question

If there is a proper class of Woodin cardinals, then Woodin showed (using stationary towers) that $(\Sigma^2_1)^{\text{uB}}$ statements are generically absolute, where $\text{uB}$ denotes the pointclass of universally Baire sets of reals. This generic absoluteness result has a more local version: if $\lambda$ is a limit of Woodin cardinals, then $(\Sigma^2_1)^{\text{uB}_\lambda}$ statements are generically absolute for posets of size less than $\lambda$, where $\text{uB}_\lambda$ denotes the pointclass of $\lambda$-universally Baire sets of reals (or what some people would call $\mathord{<}\lambda$-universally Baire sets of reals.)

The "local" generic absoluteness for $(\Sigma^2_1)^{\text{uB}_\lambda}$ can be explained in terms of trees (although the proof uses stationary towers instead.) More precisely, let $\varphi(v)$ be a formula in the language of set theory expanded by a unary predicate symbol. For every limit $\lambda$ of Woodin cardinals there is a tree $T_{\varphi,\lambda}$ such that in every generic extension $V[g]$ by a poset of size less than $\lambda$ we have

$$ V[g] \models p[T_{\varphi,\lambda}] = \{x \in \mathbb{R} : \exists A \in \text{uB}_\lambda\,(\text{HC}; \mathord{\in},A) \models \varphi[x]\}.$$

This tree is obtained from the scale property for the pointclass $\Sigma^2_1$ of the derived model of $V$ at $\lambda$. Given these trees $T_{\varphi,\lambda}$, the "local" generic absoluteness follows by a standard argument using the absoluteness of well-foundedness.

My question is, can the "global" generic absoluteness for $(\Sigma^2_1)^{\text{uB}}$ also be explained in terms of trees, assuming that there is a proper class of Woodin cardinals? More precisely, is there a single proper-class-sized tree $T_\varphi$ such that in every generic extension $V[g]$ we have

$$ V[g] \models p[T_{\varphi}] = \{x \in \mathbb{R} : \exists A \in \text{uB}\,(\text{HC}; \mathord{\in},A) \models \varphi[x]\}?$$

I can think of two possible approaches, both with apparently serious problems.

1. Consider the "derived model at $\text{Ord}$." Problem: this doesn't really exist.

2. Define $T_\varphi$ as the amalgamation of the trees $T_{\varphi,\lambda}$ for various $\lambda$, e.g. all limits of Woodin cardinals, or all limit of Woodin cardinals above some point. Problem: I don't see any way to show that the projection of such an amalgamated tree in some generic extension $V[g]$ is not too large.

Answer 1

The answer is yes. Hugh Woodin showed me the following argument, which I post here with his permission.

Let $\varphi(v)$ be a formula in the language of set theory expanded by a unary predicate symbol. Given a pair of ordinals $(\alpha, \beta)$, working in $V^{\text{Col}(\omega,\alpha)}$ we let $B$ be a universally Baire set of reals having Wadge rank $\beta$ in the model $L(B,\mathbb{R})$, which satisfies $\mathsf{AD}^+$. Note that this model depends only on $\beta$ and not on $B$, and also that every set of reals in $L(B,\mathbb{R})$ is universally Baire because $B^\sharp$ exists and is universally Baire. Let $T_{\alpha,\beta}$ be the tree of a $(\Sigma^2_1)^{L(B,\mathbb{R})}$-scale on the set $$\{x \in \mathbb{R} : \exists C \in L(B,\mathbb{R})\, (\text{HC}; \in, C) \models \varphi[x]\}.$$ By the homogeneity of $\text{Col}(\omega,\alpha)$ this tree is is independent of the choice of generic filter and we have $T_{\alpha,\beta} \in V$. Let $T$ by the amalgamation of all the trees $T_{\alpha,\beta}$, so that $T$ is a tree on $\omega \times \text{Ord}$ and $p[T] = \bigcup_{\alpha,\beta \in \text{Ord}} p[T_{\alpha,\beta}]$ in every generic extension of $V$.

We claim that $$ V^{\text{Col}(\omega,\alpha)} \models p[T] = \{x \in \mathbb{R} : \exists C \in \text{uB}\, (\text{HC}; \in, C) \models \varphi[x]\},$$ for every ordinal $\alpha$. The right-to-left inclusion follows immediately from the definition of the trees $T_{\alpha,\beta}$, so it remains to prove the left-to-right inclusion. Let $G \subset \text{Col}(\omega,\alpha)$ be a $V$-generic filter and let $x \in p[T]^{V[G]}$, say $x \in p[T_{\alpha',\beta'}]$ for ordinals $\alpha'$ and $\beta'$. We want to show
\begin{equation*}\tag{$*$} \exists C \in \text{uB}^{V[G]}\, (\text{HC}^{V[G]}; \in, C) \models \varphi[x]. \end{equation*}

If $\alpha' = \alpha$, this is easy. There are two remaining cases to consider:

1. $\alpha' > \alpha$.

2. $\alpha' < \alpha$.

In case (1), we have ($*$) by $(\Sigma^1_2)^{\text{uB}}$ generic absoluteness for $\text{Col}(\omega,\alpha')$. In case (2), we use the fact that if $B \in V[G \restriction \alpha']$ is a universally Baire set as in the definition of the tree $T_{\alpha', \beta'}$, then $B^\sharp$ exists and is universally Baire, so there is an elementary embedding $$ j : L(B, \mathbb{R}^{V[G \restriction \alpha]}) \to L(B^{V[G]}, \mathbb{R}^{V[G]}),$$ and we have $j(T_{\alpha', \beta'}) = T_{\alpha, \beta}$ where $\beta$ is the Wadge rank of $B^{V[G]}$. Considering the pointwise image of a branch witnessing $x \in p[T_{\alpha',\beta'}]$, we have $x \in p[T_{\alpha,\beta}]$ . Therefore ($*$) is witnessed by a set of reals $C \in L(B^{V[G]}, \mathbb{R}^{V[G]})$.