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

[Base-Acid Tetration]

Base-e pentation may be an entire function. [unlike tetration]

Shit, I was wrong!

[textbook speak]We will now prove that pentation is not entire:

Theorem. There exists no entire Pentation Pen_b(z), for b>1, such that Pen(0) = 1.

Proof. Let Tet_b(z) be the principal branch of tetration. Pentation (big P) satisfies Pen(z+1)=Tet(Pen(z)) We know that for z <= -2, Tet_b(z) has a branch cut, and therefore is not defined/holomorphic. Suppose, contrary to our claim, that there exists an entire function Pen_b(z), such that pen_b(z+1)= Tet_b(Pen_b(z)). By Picard's Little, pen(z), being non-constant (Pen(1) = Tet(Pen(0)) = b), surjects to the (punctured at most once) complex plane. There exist values z for which Tet(z) <= -2, e.g. a portion of the interval (-2,1] where -infinity < Tet(z) < 0; so Pen(z) must not take on these values, for if Pen(z) were to have these values Pen(z+1) would not be defined. Since there exists more than one values, in fact an interval (infinitely many) of such values in that interval, (by continuity of tetration at (-2,1] and the intermediate value theorem), which the Pentation has values in, there must be places in the complex plane on which pentation is not holomorphic. (Let z0 be such a value for which Pen(z0) <= -2. It follows that Pen(z0 + 1) = Tet(Pen(z0)) is undefined, and Pentation is not holomorphic at z0 + 1.); contradiction found. Therefore any entire Pentation must be a constant, trivial Pentation (for which Pen_b(z) is equal to a fixed point of Tet_b(z)). Halmos.
("AM I RIGHT, BO???")

It can be further proven that (1)there exists no entire non-trivial real-to-real n-exponential for n > 3; the proof is left to the reader as an exercise. [/textbook speak]

Now WHERE are teh singularities/branch points of pentation? Or alternatively we can incorporate parts of non-prinicipal branches of tetration (analytically continued around z = -2) in our construction of a holomorphic pentation? Too complex for me! [no pun intended]