The Mixed-Base Tetration Phase Map Is a Circle Diffeomorphism
Janis Justus · Zenodo (CERN European Organization for Nuclear Research) · 2026
This paper is the second part of the mixed-base tetration series and a companion to “A Phase Law for Mixed-Base Tetration.” Paper I derives exact inverse mean antisymmetry for the mixed-base tetration phase under the assumption that the associated phase map is an orientation-preserving continuously differentiable circle diffeomorphism. The present paper removes that assumption in four graded steps. First, given the phase law for both ordered pairs of bases, the lifted phase map is unconditionally an orientation-preserving circle homeomorphism whose inverse is the corresponding reverse-base phase map. The mixed-base phase maps therefore form a groupoid in the orientation-preserving homeomorphisms of the circle. Second, exact inverse mean antisymmetry already holds at this topological level through a Riemann–Stieltjes substitution, without any smoothness assumption. Third, a minimal-regularity principle shows that if both phases are continuously differentiable on the circle, the phase maps are automatically mutually inverse continuously differentiable diffeomorphisms; no separate positivity estimate is required. An explicit example demonstrates that one-sided continuous differentiability would not suffice. Finally, under an explicit continuously differentiable strengthening of the channel hypotheses from Paper I—differentiable strict monotonicity of the tetration branches and quantitative positivity of the logarithmic channel—the forward phase alone is continuously differentiable with strictly positive derivative. The proof uses an exact product identity for the derivatives of the logarithmic tails in which all tower-sized factors cancel. All statements are supported by numerical validation, including an end-to-end check using explicit continuously differentiable tetration branches.