Real-analytic tetration uniqueness criterion?

[sheldonison]

For tetration at the real axis, the nearest singularity is at x=-2, and there are no other singularities to the right of that anywhere in the complex plane.

Well that depends on what type of tetration we use.
Kneser seems to have this property.
But so do many theta variations of Kneser. ( the analytic theta's )

Conjecture; Kneser is the only solution with no singularities in the upper/lower halves of the complex plane, and this is a uniqueness criterion. For all the of the entire theta functions, we know there will be an infinite number of singularities in the upper half of the complex plane, where z+theta(z)=-2,-3,-4 ..... I would also conjecture that these singularities will be in the right half of the complex plane as well. But either way, Kneser has this special property, so Kneser's so at any value of z, there are negative even derivatives for large enough (2n). This doesn't prove that all of the odd derivatives are always positive, but that is also a conjectured uniqueness criterion.

Also why would the maximum value be at z0 ?

This is not in general true for polynomials UNLESS all derivatives are nonnegative.
But here we have some negatives. Well even that depends on parameters and conjectures.

Yes, I guess that's another conjecture, that for all real(z)>~=0.5, if you make a line from -imag infinity to +imag(infinity) at real(z), the maximum absolute value occurs at the real axis. This is also supported by empirical evidence, but I can't think of any obvious way to prove it. This would also mean that the maximum magnitude on any circle on the real axis occurs at the real axis, so long as its bigger than about 0.5.

This is related to the conjecture that all of the odd derivatives are positive; you might be able to prove one from the other. Assuming all of the odd derivatives are positive, it would be interesting to try to prove that any sexp(z+sin(z)/k) mapping has negative odd derivatives.