User blog comment:B1mb0w/Fundamental Sequences/@comment-5529393-20160124015439/@comment-1605058-20160206090532

You have never provided any justification as to why you claim phi(2,1) = phi(2,0)^^w. The equality phi(2,0)^^w =phi(1,phi(2,0)+1), whether you find it self-evident or not having any "explanatory power" whatever that is, is correct. However, if we had phi(2,1) = phi(2,0)^^w, then you would have phi(1,phi(2,0)+1)=phi(2,0)^^w=phi(2,1)=phi(1,phi(2,1)), which would imply phi(2,0)+1=phi(2,1) (because phi(1,x) is an increasing function), which I hope we agree is obviously not the case.