x↑↑x = -1

[sheldonison]

I feel like this is overdetermined.

For K = 1,2,3 : let \( \theta_K(z) \) satisfy

1) \( \theta_K(z) = \theta_K(z+1) \)

2) \( \theta_K(n) = 0 \) for every integer \( n \).

3) \( \theta'_K(n) = 0 \) for every integer \( n \).

4) \( z + \theta_K(z) \) is univalent for z near the real line.

5) \( 1 + \theta'_K(z) =/= 0 \) for z near the real line.
....

Just remember that for both theta functions, \( \theta(n) \) has a really nasty singularity at integer values of n, and is definitely not a constant. So the only place the two \( \alpha^{-1}(z+\theta(z)) \) functions agree is right at their analytic boundaries. I suppose you could probably extend the boundary at non-integers, but besides the singularities, its going to behave pretty poorly, and \( \alpha1(\text{tet}(z))-z \) is no longer 1-cyclic if \( \Im(z)<0 \) so analytic continuation isn't helpful. Also, at the real axis, even with thousands of series terms, theta(z) would still limit precision to a handful of decimal digits. So, we also avoid the real axis, and use a third function, tet(z), to glue together the two theta(z) representations, to have any hope of getting any accuracy at all for results.