Martin-Solovay Tree of Weakly Homogeneous Tree under $\mathsf{AD}_\mathbb{R}$

Question

A tree $T$ on $\omega \times \lambda$ is weakly homogeneous if there is a countable set $\sigma$ of countably complete measures on ${}^{<\omega}\lambda$ so that $x \in p[T]$ if and only if there is a countably complete tower of measures $\bar{\mu} = \langle \mu_i : i \in \omega \rangle$ so that each $T_{x\upharpoonright i} \in \mu_i$ and each $\mu_i \in \sigma$.

Fix such a tree $T$ and a $\sigma$ witnessing weak homogeneity. Fix an enumeration $\langle \mu_i : i \in \omega\rangle$ of $\sigma$ so that projections of $\mu_i$ come before $\mu_i$.

For an ordinal $\nu$, the Martin-Solovay tree $\mathrm{MS}_\nu(T,\sigma)$ is a tree on $\omega \times \nu$ defined by: $(h,s) \in \mathrm{MS}_\nu(T,\sigma)$ if and only if for all $i < l < |s|$, if $k_i = \dim(\mu_i)$ and $k_l = \dim(\mu_l)$, $T_{s \upharpoonright k_i} \in \mu_i$, $T_{s \upharpoonright k_l} \in \mu_l$, and $\mu_i$ is a projection of $\mu_l$, then $j_{i,l}(h(i)) > h(l)$, where $j_{i,l}$ is the natural map between the $\mu_i$ ultrapower and the $\mu_l$ ultrapower.

In Steel's "The Derived Model Theorem" Lemma 1.19, he shows that $p[T] = {}^\omega\omega \setminus p[MS_{\lambda^+}(T,\mu)]$. This is proved by finding a continuous witness to the ill foundedness of all appropriate towers coming from $\sigma$. At least in the Steel's proof provided there, it seems that he is using the axiom of choice.

So my question is: Under $\mathsf{AD}_\mathbb{R}$, Martin showed that every tree $T$ on $\omega \times \lambda$ where $\lambda < \Theta$ is weakly homogeneous. Fix $T$ and $\sigma$ witnessing weak homogeneity. The Martin-Solovay tree $\mathrm{MS}_{\lambda^+}(T,\sigma)$ can be constructed. Under $\mathsf{AD}_\mathbb{R}$, is $p[T] = {}^\omega\omega \setminus p[MS_{\lambda^+}(T,\sigma)]$ still true?

In Larson's Stationary tower book, he mentioned that it is unknown whether the type of continuous witnesses to illfoundedness of all towers (used in $\mathsf{AC}$ proof) can be proved in $\mathsf{ZF + DC}$.

In his Games and Scales paper, Steel mentioned that Martin-Solovay trees are studied in the choiceless $\mathsf{AD}$ context. I suspect at least under determinacy the Martin-Solovay tree should project onto the complement of a weakly homogeneous tree.

Thanks for any information or references to this question.

Answer 1

It is a theorem of $\text{ZF+DC}$ that if $T$ is a weakly homogeneous tree on $\omega\times \kappa$ some $\kappa$ with homogeneity system of measures $\vec{\mu}$ and $ms(T,\vec{\mu})$ is the Martin-Solovay tree associated to $T$ then $p[T]$ and $p[ms(T,\vec{\mu})]$ are complements. See Cabal reprints volume 1 Jackson's article theorem 4.10 for a proof. Note that to show weak homogeneity of trees on $\omega\times \kappa$, where $\kappa$ less than the supremum of the Suslin cardinals, one actually only needs $\text{AD}$ (result of Woodin).

$\text{AD}$ or $\text{AD}_{\mathbb{R}}$ are used to show homogeneity or weak homogeneity of trees. This is done using partition properties in the determinacy context (see Jackson's article mentionned above). These assumptions are not needed to establish that a weakly homogeneous tree $T$ and its Martin-Solovay twin project to complements (if that's what you were looking for).