I've only recently learned about Girard's theory of Dilators and Ptykes, and I find this theory very elegant, but it is not clear at all to me whether/how it can be used to produce ordinal notations for all the large recursive ordinals used in proof theory and ordinal analysis. The introduction of several papers on the topic seems to claim it is possible - but I can't find it done anywhere...
I think I can see how Dilators alone can be used to construct (something very similar to) the Veblen functions in (possibly infinitely) many variables and how to get ordinal notations up to maybe the large Veblen ordinal, or a little higher than this.
However - and I have no idea how to make this formal - these constructions feel very "predicative" and my intuition would be that there is some kind of limit to what we can build using these only, I would guess around the Bachmann-Howard ordinal... But maybe I'm wrong and there is a way to formalize something similar to the ordinal collapsing functions using this theory?
General Ptykes on the other hand, feel much more mysterious to me, and I'm not sure how they can be used - I find the literature on the topic doesn't provide many examples - but they do seem more powerful, so I wouldn't be surprised if they could go as high (and probably higher) than everything we get using ordinal collapsing functions... But I don't know how.
Basically, I'd be interested by any reference or answer that gives Dilator/Ptykes based description of Ordinals notations up to and above the Bachman-Howard ordinal. I'd be also interested in results that gives limitations to such methods...