Interesting, so we'd have branches of \( \text{slog} \) that are holomorphic in a neighborhood of \( L \) then. Unlike with the logarithm where no branch is holomorphic in a neighborhood of \( 0 \). I think the trouble with this construction will be getting \( \Phi \) to be well behaved as we grow \( \Re(s) \). And not to mention, getting general growth lemmas on the various branches of \( \text{slog} \). The only work around I see would be to take the inverse iteration,
\(
\text{pent} = \text{tet}_{\text{Kneser}}^{\circ n} \Phi(s-n)\\
\)
Where of course, we'd have to modify \( \Phi \) to converge in this circumstance. And here we'd probably lose any chance of it being real-valued. I'm going to keep this on a backburner and come back to it later. I'm going to stay focused on \( \mathcal{C}^\infty \) proofs for the moment.
\(
\text{pent} = \text{tet}_{\text{Kneser}}^{\circ n} \Phi(s-n)\\
\)
Where of course, we'd have to modify \( \Phi \) to converge in this circumstance. And here we'd probably lose any chance of it being real-valued. I'm going to keep this on a backburner and come back to it later. I'm going to stay focused on \( \mathcal{C}^\infty \) proofs for the moment.