Universal uniqueness criterion?

[Base-Acid Tetration]

Yes exactly. If you have one super-logarithm that maps G to *some* region and another super-logarithm that maps G to *some other* region , and both super-logarithms satisfies the other conditions of the proposition, then they must be equal. Hence this assumption of different images was already wrong: the images are equal .

Can you reproduce the proof here?

Not sure why you think it wouldnt work. The only problem with our tetration methods may be the biholomorphy or merely the holomorphy.

Because your paper said there is only one tetration/superlogarithm that maps g to SOME strip. I missed the beginning of the proof, that f and g were assumed to map G to different strips.

EDIT; ok, you proved that f and g really map G and the strip equally, so they are the same holomorphic function.
Cool, now we have to prove the biholomorphicity of the different tetration methods between G and the strip, and according to the theorem, since they are all tetration/tetralogarithm (superfunction/Abel function of exponential), they are all the same. Does the applicabilty of theorem 1 only require that the function F (in the case of tetration, exp) is holomorphic, and F (exp_b, b>e^1/e) has no real fixed points?
Then what are you guys griping about in threads like Bummer!?