The ultimate sanity check

[bo198214]

My second concern is that proofs based on Kneser's paper may well begin satisfying \(f^{a+b}(z)=f^a(f^b(z))\), but I question whether the identity survives the mapping to the unit circle and then into the real line.

Yes, it survives the mapping. I made a sequence of posts explaining the Kneser-method, you would find the explanation here. The essential thing is that the region \(L\) that is mapped consists of repeating stripes and hence L+c=L. Hence the mapping \(\gamma=\beta\circ\alpha\) to the upper half plane can be choosen to be \(\gamma(z+c)=\gamma(z)+1\) which in turn make the "pre-Abel" function \(\psi(\exp(z))=\psi(z)+c\) into an Abel function \(\Psi=\gamma\circ\psi\).