Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation

[Base-Acid Tetration]

Tetration isn't entire, it's holomorphic over the complex plane cut along the negative reals < -2.

I said PENTation.

Theorem. There exists no entire pentation pen_b(z), for b>1, such that pen(0) = 1.

Either you have to specify that pen must use the principal branch of tetration (the cut is nothing god given, its just a choice to our human preference; the real function lives on Riemann manifolds), or you have to modify your proof and use the singularities of tetration at integers <=-2 (and show before that indeed every tetration has singularities there).

I DID say:

Let Tet_b(z) be the principal branch of tetration.

I fixed the proof to make this clearer.

I also did say that if we didn't restrict ourselves to the principal branch an entire pentation may be possible:

Or alternatively we can incorporate parts of non-prinicipal branches of tetration (analytically continued around z = -2) in our construction of a holomorphic pentation?

I meant if it can just avoid -2, pentation can still be entire.

Now the question is, is it morally good to restrict ourselves to the principal branch?

(BTW, what kind of singularities does Tet have at integers<=-2? Are they all logarithmic-type branch points around which the function winds infinitely many times?)

Let's get back to the point. WHERE THE HECK ARE THE COMPLEX FIXED POINTS?