Circulation and the Fast-Growing Hierarchy

[Catullus]

Unless I'm the one confused here, it might be, I suggest to NOT confuse circulation/omegation with FGHs.

The omega in omegation/circulation is NOT an ordinal. It is just notation sugar, it has nothing to do with the ordinal omega. Also the operation \(o(a,b)=\lim a\uparrow^n b\) is not even well defined, let alone taking superfunctions out of it.

Also the point of FGH is that they extend Ackermann-Goodstein \(a\uparrow^n b\) like functions from \(n\in\mathbb N\) to \(n\in {\bf On}\), but not to all the ordinals but only to sufficiently small ones, i.e. transfinite ordinal that must be countable \(|\alpha|\leq\aleph_0\), and are also recursively definable in some technical sense (or you can compute the fundamental sequences).

Also there is not a single way to extend it to transfinite ordinals, but multiple ways to do it, some more natural than others, and all the various ways depend fundamentally on a choice of a system fundamental sequences. A fundamental sequence is a system of choices about how to define it for limit ordinals, and to my limited knowledge, it amounts to an algorithm of diagonalization. In other words \(a\uparrow^\omega b\) has not really something to do with the idea of infinity or limit but it is defined using a trick, something like defining \(a\uparrow^\omega b=a\uparrow^b b\).

Why is the operation a circulated to the b not well defined? Two circulated to the two still equals four. Two circulated to the five may be some large infinity. Still defined.
Not all of the levels of the Fast-growing hierarchy are recursive. Not all countable ordinals have computable fundamental sequences even if they recursive ordinals. The Church-Kleene ordinal (https://googology.fandom.com/wiki/Church-Kleene_ordinal) for instance, is non recursive and non computable.
It might be interesting to extend the hyper-operation hierarchy to transfinite ordinal ranks.
I wonder how omega minus one would behave in some sort of extension of the Fast-growing hierarchy. I know omega minus one is not an ordinal. But it is a Hyperreal number.
would a be a sequence, such that when applied x time to x it results in . What about extending the Fast-growing hierarchy to other Hyperreals? such as omega minus 2?