Are tetrations fixed points analytic?

[JmsNxn]

Algebraically we can characterize \( \tau_\alpha \) by the equations

Yes it is analytic. There is the well known equation for the fixed point using the LambertW function.
\( \alpha^{\tau_\alpha}={\tau_\alpha} \;\;\;\; {\tau_\alpha} =\frac{W(-\ln(\alpha))}{-\ln(\alpha)} \)

Are you talking about exps fixed points, or tetration's fixed points? You wrote exp, and I'm pretty sure that's the equation for exps fixed points. I'm interested in, \( ^{\tau_\alpha} \alpha = \tau_\alpha \) not \( \alpha^{\tau_\alpha} = \tau_\alpha \). I'm well aware exps fixed points are analytic, I'm interested in the fact that if we keep on increasing the hyper operator index in the bounded case that the hyper operators always have an analytic fixed point function and hence geometrically attracting fixed points.