Universal uniqueness criterion?

[Base-Acid Tetration]

(3) We can repeat (4.2) as many times as needed to get to \( D_2 \).

No, we can not.
1. The curve may get out of the strip of bijectivity of what indeed happens for . (plot the curves!)
2. may hit , i.e. the extension of overlaps with the original domain. If meets at an angle then meets at an angle . So the image curves rotate around L. This includes also the comment of Ben, that if , i.e. if is real, then then the images do not rotate, and hence one would never reach if it hits in a different angle.

Yeah, I knew it wouldn't work right when I postd it, because of the issues with f's domain of biholomorphy. I edited that many times, but exp still doesn't meet the condition.

Such a function should have some smooth kink of the phase, in order to avoid the jump; but allow such a jump for the principal branch of its logarithm...
I think about something like \( E(y)=
\Re(L_1) +(1-\Re(L_1))/ \cosh(y) + \Im(L_1) \tanh(y) \)

@Kouznetsov: Can you show me how E(y) would have the behavior as you speculate? Also do the tetrations constructed with other fixed point pairs still satisfy the basic functional equations b[4](z+1) = b^b[4]z and b[4]1=b?