User blog comment:Edwin Shade/A Proof/@comment-1605058-20171104122735

Let me address one thing you had in the post before removing it, namely the definition of the fractional hyperexponentiation: if you want it to only depend on the rational number \(\frac{p}{q}\) and not on \(p,q\) themselves (which it has to if you want to extend it to real numbers later), then that definition simply doesn't work: note that we have \(2\Delta_4\frac{2}{1}=(2\Delta_4 2)\nabla_4 1=4\nabla_4 1=4\), whereas \(2\Delta_4\frac{4}{2}=(2\Delta_4 4)\nabla_4 2=65536\nabla_4 2\approx 6.121\dots\).