User blog comment:Mh314159/YIP notation/@comment-35470197-20190704230823/@comment-35470197-20190705005758

Please never mind. This kind of attempts is really helpful for you to undestand large functions and FGH. To begin with, please remember my lower most comment here explaining the difference of the recursion in FGH and your systems. Follwing the explanation, FGH is a system which generate a new function by iterating the srongest function ever defined. On the other hand, your recent function [a,3] does not iterate the strongest function [a,2] but iterates a weaker function [a,1] ([a,2] times).

For large functions \(f(n)\) and \(g(n)\) such that \(f(n)\) is significantly larger than \(g(n)\), the iteration \(f^n(n)\) of \(f\) is much stronger than the iteration \(g^{f(n)}(n)\) or even \(g^{g^{\cdot^{\cdot^{\cdot}}}(n)}(n)\) (\(f(n)\) \(g\)'s), although the latter ones look so complicated. Once you clearly understand this relation, then you will soon go beyond \(\omega^{\omega}\). (Then you will find a big wall at \(\omega^{\omega^{\omega}}\). You will need another knowledge on recursive notations here.)

I really enjoy your effort, because you are pretty good at formalising. Even if a googologist knows a system beyond TREE(3), he or she usually does not possess the ability to write down precise definitions in their mind. On the other hand, you can display what you considered so clearly. It is greatly helpful for yourself to study and cultivate new ideas.