eta as branchpoint of tetrational

[bo198214]

I see now, you're talking about in the base. I was referring to in the height Never mind, just missed that.

But anyway, wouldn't showing that it was not conjugate-symmetric in the height when \( b < \eta \) work?

And it seems we both confused to a certain extend base and height.

Ok, I see your (two) points now:
Given that it converges in the upper halfplane to the upper fixpoint and correspondingly in the lower halfplane, then
1. Its not conjugate symmetric when approaching the real axis below eta, and hence can not be real valued.
2. it approaches a different function from above and below

Yes, this seems not to contradict the conjugacy when approaching from above and below.

So we somehow need to verify your assumption, that it converges to the upper fixpoint for positive imaginary infinity. Perhaps I will write a Wiki article about perturbed Fatou coordinates, which may help to decide this question.