A version of the Martin–Solovay tree for $H_\kappa$ Consider a fixed $\Pi^1_2$ property of reals, $A(x)$.
Is it true that relative to a regular cardinal $\kappa$, one can define a version of the Martin–Solovay tree $T_2$ for $A$ with the following properties?


*

*$T_2$ is defined under the assumption of ZFC + "every set in $H_\kappa$ has a sharp."

*$T_2$ is a tree on $\omega \times u_\omega$, where for every ordinal $\alpha \ge 1$, $u_\alpha$ is the $\alpha^\text{th}$ uniform indiscernible for the models $L[x]$, $x \in H_\kappa$. (So $u_1 = \kappa$ and $u_2 \le \kappa^+$.)

*$T_2$ projects to the set of reals $\{x \in \omega^\omega : A(x)\}$.

*For every inner model $M$ of ZFC satisfying "every set in $H_\kappa$ has a sharp," if $u_2^M = u_2$ then $T_2^M = T_2$.


If so, where can I find a reference for this? I have only been able to find two special cases.
The most well-known case seems to be $\kappa = \omega_1$, in which case "every set in $H_\kappa$ has a sharp" simply means "every real has a sharp" and a proof of property 4 can be found in Hjorth, The size of the ordinal $u_2$, Theorem 2.1.
The case where $\kappa$ is a measurable cardinal can be found in Steel, The Core Model Iterability Problem, Section 7D. The argument there refers to $V_\kappa$ instead of $H_\kappa$, which of course is equivalent in this case. Property 4 is proved in Lemma 7.10.  
Both Hjorth and Steel credit their arguments for property 4 to Magidor but do not cite any written work of his.
I think a common generalization like I stated above should hold by a similar argument.  Perhaps that is what Magidor actually proved?  I would be interested in any mention of the general case in the literature, since I have not been able to find any.  (Or if the generalization I stated is invalid for some reason, of course I would like to know that too.)
 A: Assume $AD+V=L(\mathbb{R})$. Let $\kappa\leq\delta^2_1$ be a regular reliable cardinal (for example a projective ordinal of the form $\delta^1_{2n+1}$) and let $u_{\alpha}^{(\kappa)}$ be the $\kappa^{th}$ uniform indiscernible for subsets of $\kappa$, that is $u_1^{(\kappa)}=\kappa^+$. Recall that $H_{\kappa^+}$ can be coded by $\mathcal{P}(\kappa)$. Assume that $\mathcal{P}(\kappa)^{\#}$ exists. We will simplify the set up and assume that $\kappa=\delta^1_{2n+1}$ (or in general, we may assume that $\kappa$ is the Suslin cardinal associated to a pointclass $\Pi$ which is $\Pi^1_1$-like.
We wish to look at $u_{\omega}^{(\kappa)}$. Let $\nu$ the a supercompactness measure on $\kappa$ (this follows from work of Becker and Jackson). We need to first compute $u_{\omega}^{(\kappa)}=j_{\nu}(\aleph_{\omega})$ (this last equality is precisely theorem 7.2 in the article Generic Codes of Kechris-Woodin using the continuity of the supercompactness of the elementary embedding. 
By work of Jackson and the above embedding, we must then have that $u_{\omega}^{(\kappa)}\leq \aleph_{\omega^{\omega^{\omega^{...}}}+3}$, where the towers of $\omega$'s has height $2n-1$ (this is where we use the simplification we made, this analysis is only true for an initial segment of $L(\mathbb{R})$, how far it goes is unknown).
Finally we can define the version of the Martin-Solovay tree for $H_{\kappa^+}$ as follows: Let $T^2(\kappa^+)$ be a tree on $\omega\times u_{\omega}^{(\kappa)}$ in the usual way and compute its projection.
(This answer is temporary, feel free to add to it, I will revisit it and add more details and corrections, for now the subscripts are confusing me I would need $A$ to be a $\Pi^1_2(B)$ property for some suitable set $B\subseteq \mathbb{R}$, maybe one could take $B$ a universal $\Pi$-set of reals where $\Pi$ is the $\Pi^1_1$-like pointclass of above. In addition, I believe my subscripts are off because to compute the Martin-Solovay tree we would need a suitable measure. Finally property 4) above would follow from theorem 4.3 in Jackson's "The Weak Square Property". In any case, sorry for the half-non-answer, this need to be completed although this seems to be the right direction).
