Universal uniqueness criterion?

[bo198214]

"There exists only one holomorphic superlogarithm with b>e^1/e that maps G biholomorphically to *some* region with upper and lower unbounded imaginary part"

That must mean that we still don't know WHAT region the superlogarithm must map G (the crescent moon thingy) to. So our different tetration methods might map G to different infinitely long strips of width 1.

Yes exactly. If you have one super-logarithm that maps G to *some* region \( H_1 \) and another super-logarithm that maps G to *some other* region \( H_2 \), 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 \( H_1=H_2 \).

Not sure why you think it wouldnt work. The only problem with our tetration methods may be the biholomorphy, the bijectivity of the super-logarithm on the initial region (or merely the holomorphy for some methods).