Universal uniqueness criterion?

[Base-Acid Tetration]

I think if both initial regions belong to the same fixed point pair, then both slogs are analytic continuations of each other, i.e. basically the same holomorphic function. But this is not yet proven.

I'll try to prove that this summer. just give me all the relevant facts.

two Abel functions biholomorphic on two different initial regions

two Abel functions: A1(f1(x)) = A1(x) + 1 , A2(f2(x)) = A2(x) + 1
you mean two different abel functions, (1) where f1, f2 are different functions? or do A1, A2 map the same point to different values? (for sake of argument, a "superlogarithm" that does slog(0) = 0)