Status of proofs

[bo198214]

I am interested to hear what people consider needs to be proven to establish tetration for real and complex numbers.

Well, for me the following assertions need to be proved.

1. Do the coefficients in Andrew's solution converge?
   
2. Do the series made up of this coefficients converge? What is the convergence radius?
   
3. Does the natural development at every point define the same analytic function? Is the piecewise defined function analytic?
   
4. Is Andrew's solution for bases\( b\le e^{1/e} \) identical with the solution obtained by regular development at the lower fixed point?
   
5. Is the solution analytic with respect to the base as argument (especially at the base \( b=e^{1/e} \))?
   

I think each question can be answered with a yes (and there are strong numerical hints for it), however each proof is at least not obvious.
I call it Andrew's solution merely to give it a common name, though the idea was published before him by Peter Walker.