It appears you have already beat me to the chase with
\[
\phi(t) = \frac{\Phi^t - \cos(\pi t)|\Psi|^t}{\Phi-\Psi}\\
\]
Which has the exact same shape as I wrote above.
The real question is uniqueness then! I think we'll get uncountably infinite real super functions. Jesus!
\[
\phi(t) = \frac{\Phi^t - \cos(\pi t)|\Psi|^t}{\Phi-\Psi}\\
\]
Which has the exact same shape as I wrote above.
The real question is uniqueness then! I think we'll get uncountably infinite real super functions. Jesus!

