Call $X$ *very hyperlow* if $\mathcal{O}^X \le\_T \mathcal{O}$, where $\mathcal{O}$ is your favorite $\Pi^1\_1$-complete set. Note: Turing reducibility, not hyp-reducibility. Observe that this is a (Turing) degree invariant notion.


Are the very hyperlow Turing degrees closed under join?



Are the very hyperlows closed under join?
I don't think so. There is an $\mathscr{O}$-recursive sequence $\{D\_i\}\_{i}$ of dense open sets so that for any real $g$ meeting every member of the sequence, $\mathscr{O}^g\leq\_T g\oplus \mathscr{O}$. Now it is simple to construct two such reals $g\_1$ and $g\_2$ so that $g\_1\oplus g\_2\equiv\_T \mathscr{O}$.


