Real and complex tetration

[Ember Edison]

WARNING: Religious topic discussed!!!

Is real tetration dependent on complex tetration? Could real tetration exist without complex tetration?

Very difficult. There exists Smooth real tetration; but it isn't analytic--and there are uncountably many of these. The trouble is, we can also make uncountably many real analytic tetrations by just adding a well behaved enough \(\theta\) mapping. By which; these wouldn't succumb to the complex uniqueness conditions though.

So No, real tetration is not dependent on complex tetration. It's just that complex tetration allows us the only confirmed construction--and allows us the uniqueness conditions we so love.

I want to confirm something.

1. Is \( R^\infty \) not enough to uniquely determine an real tetration?
2. Are there two (mainstream) real tetration sexp(base, height) defined in two different ways, one of which satisfies \( e^{-e} \leq b \leq e^{\frac{1}{e}} \) and the other satisfies \( b > 1 \), and the natural extension of the latter is Kneser?