Arguments for the beta method not being Kneser's method

[JmsNxn]

I'm so sad. Why holomorphic can’t uniquely determine Tetration?
This is like to when you spend a lot of time solving the equation and finally find that the equation is underdetermined.

Honestly it makes sense to me. Kneser's solution is characterized by the Schroder functions about \( L,L^* \); and by a decay to these constants at \( \Re(z) = -\infty \) and \( |\Im(z)| \to \infty \). Really, you can think of Kneser's solution as the only real valued iteration about the fixed points \( L,L^* \); so there is uniqueness. The trouble is these solutions are highly non-unique on the real line; even just throwing a periodic function in the mix can wig things up. But, I mean, why not just pick a fixed point out of a hat? That's how I sort of feel about Kneser.

I believe William Paulsen and Samuel Cogwill's uniqueness condition is quite beautiful (Which is largely just based around Henryk and Dmitrii's work). Personally though; I'm very opposed to using fixed points, it just feels unnatural to me--sort of, arbitrary; like "why that fixed point and not this one?". "A kind of" Fixed point at infinity seems a bit more natural to me. Also, tetration diverging to infinity as we increase the imaginary argument also seems more anomalous--I think it represents well just how whacky tetration is. If anything, I've just thrown a wrench in the gears; but I think it's a good thing. Have you seen what proposed pentations/hexations/septations look like with kneser?--they look less than desirable.

I think at this point, in the quest for "the right tetration"; which ever one runs faster and simpler and solves the storage of large numbers in a better way will probably win out. It's definitely Kneser's at the moment. I still feel Kneser is the superior tetration, simply because it's much better behaved, and taylor series are much easier to grab. I'm still having trouble making a non glitching program. God damn overflow errors. Need a perfect turing machine with geometric convergence speeds   .

Regards, James