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.

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

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.

contained inthe desired $(\Sigma^2_1)^{\text{uB}_\lambda}$ set. I have no idea whether this is true. (The analogous containment is true if you consider various sizes of Shoenfield tree for $\Sigma^1_2$, so maybe there is hope.) $\endgroup$