Consider the following statement (which follows easily from various results found in the literature):
(†) There exists a primitive recursive (“p.r.”) relation $T$ on the ordinals such that, if $(f_e)_{e<\omega}$ is a standard numbering of the p.r. functions of one variable on the ordinals, we have $f_e(x)=y$ iff $\exists z.T(e,x,y,z)$; moreover, there then exists such a $z$ which is less than the smallest p.r.-closed ordinal $\geq x$.
One straightforward consequence of (†) is a Kleene normal form theorem for $\alpha$-recursion: there exists a p.r. relation $T'$ on the ordinals such that, if $(\varphi_e)_{e<\omega}$ is a standard numbering of the partial recursive functions of one variable on the admissible ordinal $\alpha$, we have $\varphi_e(x)\simeq y$ iff $\exists z<\alpha.T'(e,x,y,z)$. (One straightforward consequence of that is that there exists a “universal” partial recursive $g$ on $\alpha$ with the property that $g(e,x) \simeq \varphi_e(x)$.)
Now every proof of (†) that I was able to find boils down to something like this: if $f_e(x)=y$ then for $\beta$ large enough (larger than all the intermediate computation values) we have $L_\beta \models f_e(x)=y$, and “$L_\beta \models f_e(x)=y$” is a p.r. relation of the ordinals $e,x,y,\beta$ because it is a p.r. relation of the sets $e,x,y,\beta$ and p.r. relations on ordinals and on sets that happen to be ordinals coincide.
This proof is explicit, in the sense that it actually gives $T$, but the $T$ in question seems rather insanely complicated (as a p.r. relation on ordinals) because of the process of converting a p.r. relation on sets to one on ordinals and because of the route through formulas and the $L_\beta$.
Question: Can the statement (†) above be proved entirely at the level of p.r. functions and relations on the ordinals? Variant: Can we give an explicit $T$ that is reasonably short?
In the case of ordinary recursion, it is not too difficult. What I would like to understand is whether there is some reason why the transfinite ordinal case should be harder or whether it is just an artefact of the way things are written in the literature.
Motivation: Partially from trying to get a better understanding of this question I asked recently; partially from thinking about how, if $\alpha$-recursion is viewed as a transfinite programming language, one would proceed to write an “interpreter” for the language in the language itself (=universal function).
