x↑↑x = -1

[sheldonison]

\( \alpha_1(\text{tet}_b(z)) = z + \theta_1(z) \)

\( \alpha_2(\text{tet}_b(z)) = z + \theta_2(z) \)
....
\( \text{tet}_b(z) = G^{[-1]}( \G(\text{tet}_b(z-1)) ) \)

Hence because G^[-1](G(z)) = id(z) is absurd we get ( by branches )

one of the following potential conclusions.

1) G^[-1](G(z)) =/= b^z

And this implies that \( \text{tet}_b(z) \) is a superfunction of 2 functions !!??

2) G^[-1](G(z)) = b^z or G^[-1](G(z)) = b^z + 2pi i / ln(b)

Very unlikely.

Since both potential conclusions are very very likely wrong , this implies

\( \alpha_1(\text{tet}_b(z)) = z + \theta_1(z) \)

\( \alpha_2(\text{tet}_b(z)) = z + \theta_2(z) \)

has no solution.
....
But then what did mike and sheldon compute !??

Why would your logic apply only to complex bases? Your logic should apply equally to Kneser's real valued tetration which also has two Abel functions, with different periodicities, etc, for L and L*. In fact, if I plug in a real valued base to tetcomplex, it calculates an Abel function and theta mapping for each fixed point, ignorant of the fact that they are complex conjugates of each other. By symmetry, the solution is a real valued Schwarz reflection, and is the same as Kneser's solution.

Later, I will find the flaw in the logic of this contradiction, but Tommy will probably figure it out first.