I am currently trying to invent a new method to extend transfinitary Extended Buchholz's function compatible 四関数, but my attempt does not seem to be completed soon. In order to make my idea clearer for myself, I summarise my thoughts.
It is good to first fix a correspondence from the hierarchy of transfinitary varible for 四関数 to a hierarchy of non-recursive ordinals. It is because I guess that the collapsing requires the mutual recursion of the OCF, the closure function, and the function for a hierarchy. If I can grasp the structure of the corresponding hierarchy, it helps me to define an OCF. (Here, I am talking about how to define an OCF. I do not intend to justify wrong analyses based on inttuitive correspondence to non-recursive ordinals without any specific OCF. See Common failure 8.1 Intuitive Analysis by Non-recursive Ordinals.)
I recall how 四関数 is intended to recursively imitate the iteration of "transfinitarisation" in the collapsification process of an OCF, together with histrical backgrounds in googology. For more details on collapsification, see User blog:p進大好きbot/Study of side nesting from the view point of collapsification.
Base layer[]
The base layer in the hierarchy of transfinitary variable is the class of pairs \((\alpha,\beta)\) of two ordinals. Therefore an OCF which uses the base layer as an input is just a \(2\)-ary OCF.
Denis defined a \(2\)-ary OCF called Extended Buchholz's function, and I think that it is the first OCF in googological communities. Later, I created an ordinal notation associated to Extended Buchholz's function, and I think that it is the first ordinal notation associated to an OCF.
Second layer[]
The second layer in the hierarchy of transfinitary variable is the class of transfinitary data of ordinals, which are naturally implemented as a finite set of pairs of variables.
For example, the pair \((\alpha,\beta)\) of two ordinals in the base layer can be identified with the transfinitary datum \(\{(1,\alpha),(0,\beta)\}\), and the tuple \((\alpha,\beta,\gamma)\) of three ordinals can be identified with the transfinitary datum \(\{(2,\alpha),(1,\beta),(0,\gamma)\}\).
Kanrokoti extended my ordinal notation to KumaKuma ψ functions, which are \(3\)-ary extenions and \(4\)-ary extensions, and I defined a transfinitary Extended Buchholz's function, which is intended to be compatible with KumaKuma ψ functions. Later, I created an ordinal notation associated to transfinitary Extended Buchholz's function.
Elements \((a,\xi)\) of transfinitery data roughly correspond to the hierarchy of inaccessibility. For example, the element \((1,\alpha)\) of the transfinitary datum \(\{(1,\alpha),(0,\beta)\}\) corresponding to the pair \((\alpha,\beta)\) of two ordinals "indicates" the class of ordinals which is \(0\)-inaccessible or a limit of \(0\)-inaccessible cardinables, i.e. the class of uncountable cardinals.
Similarly, the element \((2,\alpha)\) in the transfinitary datum \(\{(2,\alpha),(1,\beta),(0,\gamma)\}\) corresponding to the tuple \((\alpha,\beta,\gamma)\) of three ordinals "indicates" the class of ordinals which is \(1\)-inaccessible or a limit of \(1\)-inaccessible cardinables, i.e. the class of inaccessible cardinals and limits of inaccessible cardinals.
Although the intuitive correspondence to non-recursive ordinals does not provide a proof of analysis, we have reasonable guess that \(3\)-ary KumaKuma function is expected to be comparable to Rathjen's OCF based on the least weakly Mahlo cardinal up to weakly inaccessible cardinals, \(4\)-ary KumaKuma function is believed to be comparable to Rathjen's OCF based on the least weakly Mahlo cardinal up to weakly \(2\)-inaccessible cardinals, and transfinitary Extended Buchholz's function is believed to be comparable with Rathjen's OCF based on the least weakly Mahlo cardinal up to \(\psi_{\chi_M(0)}(0)\).
Third layer[]
The third layer in the hierarchy of transfinitary variable is the class of a finite set of data in the second layer. Why?
A transfinitary datum of ordinals is realised as a finite set of pairs of ordinals, and pairs of ordinals itself is regarded as a transfinitary datum of ordinals. Therefore transfinitary datum of ordinals is regarded as a finite set of transfinitary data, i.e. a finite set of data in the second layer.
In this way, a datum in the second layer is naturally regarded as a datum in the third layer. The first entry \(a\) of an element \((a,\xi)\) in a datum in the second layer appears in the second entry of the element \((1,a)\) in the corresponding daturm in the third layer.
This means that the element \((1,a)\) of a datum in the third layer "indicates" the class of \(a\)-inaccessible cardinals and limits of \(a\)-inaccessible cardinals.
Similarly, the element \((2,b)\) of a datum in the third layer "indicates" the class of \(b\)-Mahlo cardinals and "limits" of \(b\)-Mahlo cardinals. The reason why I double-quoted "limits" is because the actual role depends on whether the datum includes \((1,a)\) for some \(a \in \textrm{On}\) or not.
If the datum does not have such an element, then it actually "indicates" the class of \(b\)-Mahlo cardinals and "limits" of \(b\)-Mahlo cardinals. On the other hand, if \((1,a)\) is an element of the datum for some \(a \in \textrm{On}\), then it "indicates" the class of \(b\)-Mahlo cardinals and \(a\)-inaccessible limits of \(b\)-Mahlo cardinals. (In addition, I expect that the treatment of Mahlo cardinals which are limits of Mahlo cardinals is similar to that of Rathjen's OCF based on the least weakly compact cardinal.)
Although there is no actual comparison to Rathjen's OCF based on the least weakly compact cardinal, I expect that elements \((3,\xi)\) of transfinitery data in the third layer roughly correspond to the hierarchy of weak compactness, i.e. \(\Pi^1_1\)-indescribability.
Further, I expect that elements \((a,\xi)\) of transfinitery data in the third layer roughly "indicate" the hierarchy of shrewdness from a very rough and intuitive observation that the recursion structure of the third layer for the case \(a \in \omega \setminus \{0,1,2\}\) looks very similar to that of \(\Pi^1_{a-2}\)-indescribability in Arai's OCF based on indescribability.
Other layers[]
For any \(n \in \omega\), the \((1+n)\)-th layer in the hierarchy of transfinitary variable is the class of a finite set of data in the \(n\)-th layer by an obvious reason.
What hirerarcy do elements of a datum in the fourth layer "indicate"? \(2\)-shrewedness? I have no idea.