*The context for this question is the theory ZFC + a measurable cardinal, although answers not in this context would also be interesting to me.*

In a project I'm working on, the following class of reals has emerged, and I'd like to understand it better:

Say that a real $r$ is **hyperarithmetic modulo ordinals** (and write $r\in\Delta_1^{1, ord}$) if $r$ can be defined in a $\Delta^1_1$ way relative to some ordinal parameters. Formally, $r$ is $\Delta_1^{1, ord}$ if there is a tuple $\overline{\alpha}$ of ordinals and a pair of formulas $\varphi(x, \overline{y}), \psi(x, \overline{y})\in\Sigma^1_1$ in only the displayed parameters such that, whenever $\overline{c}$ is a tuple of reals coding copies of $\overline{\alpha}$, we have $$r=\{n: \varphi(n, \overline{c})\}=\{n: \neg\psi(n, \overline{c})\}.$$

Now, this class of reals is *much* bigger than $\Delta^1_1$. For example, Kleene's $\mathcal{O}$ is $\Delta_1^{1, ord}$: $\Phi_e$ is well-founded iff it embeds into $\omega_1^{CK}$.

EDIT: as soon as I posted this, I realized that this can be pushed further: unless I'm missing something, every constructible real is $\Delta_1^{1, ord}$. So the right question now is:

Is $\Delta_1^{1, ord}=L\cap\mathbb{R}$?

Currently I suspect the answer is "yes" - note that $\Delta_1^{1, ord}$ is forcing absolute! - but I don't see how to prove it.

A more conservative question is:

Is there a $\Delta^1_3$ real which is not $\Delta^{1, ord}_1$?

In general, any information about this class (and its obvious variations - e.g. $\Sigma^{1, ord}_n$) would be valuable to me. I suspect this is all very well-known in the descriptive set-theory community, so I've added the "reference-request" tag.

independentlyof what codes we pick, so this isn't a problem. $\endgroup$everyordinal $\alpha$ has some copy $c$ such that the $\theta$th jump of $c$ doesn't compute $x$. This was (basically) proved by Richter. $\endgroup$1more comment