Generalized Kneser superfunction trick (the iterated limit definition)

[JmsNxn]

PROGRESS UPDATE (Feb 05, 2021):
I'm making a lot of progress on point 4 but the road is way longer then I believed. I must delay a post on this for another week at least. I need time to check all the details and I'm convinced that I can go very far in generalizing the trick even more. Now I'm going from the special cases to the more general, I'm adding many pages and correcting many typos and phrasing.

Here is a spoiler of where I am currently: I can now, 99%, take on the "Nixon's trick" at the same level of generality I treated the "Superfunction trick". That's not much but it's promising. Here's the vague statement.

What you do in fact is to take \( \chi(\sigma):=\chi(\sigma,0) \), for \( h_j(\sigma,x)=e^{\sigma-j}\cdot F(x) \) and \( f \) the successor,  and to plug  \( \chi(\sigma) \) in the classical superfunction trick to obtain a supefunction of F. (apologize for mixing up letters and notation, I'll work on uniformity later).
The plan is to prove all the algebraic stuff and then using Jmsn's theorems as black-boxes to derive as corollaries something useful.

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For this reason I'll take more time for this and turn to the point 3), aka the "superfunction complete spaces".

Yes, that is exactly what I'm driving at. I think in the space,

\(
Q = \{f \in \mathcal{C}^1(\mathbb{R}^+ \to \mathbb{R}^+); f\,\text{ is a bijection}, f' \neq 0\}
\)

This method definitely works with the exponential convergents. In the general case you definitely have to be more subtle. Where in this space, for all \( f,g \in Q \) we can always find \( \phi \in Q \) where, \( f\phi=\phi g \).


EDIT:

The trouble I see with this space is that the super function of \( f\in Q \) will not exist here \( F\not\in Q \). I don't envy you if you're trying to create a general structure to where the superfunction sits.