UnAbel to do that

[sheldonison]

Just to reiterate my position (har har har), I study dynamics. So when I discuss tetration, it is solely as an example of an iterated function.... 
All systems in physics are measure-preserving dynamical systems. So any physical tetration system will have a base on the Shell-Thron boundary curve. Now strap yourselves in. The complex Shell-Thron boundary curve is associated with multipliers of the roots of unity...

No matter what approach you take, studying tetration involves mastering a lot of complex dynamics. Your approach seems like a personal taste to focus strictly on the Schroder equation and on Ecalle's Abel function solution for f^[n] for rational roots of unity with denominator of "n", for bases on the Shell Thron boundary.

I'm not fluent in the math behind graduate level Physics, but the focus on neutral fixed points seems really cool.