User blog comment:Chronolegends/Chronolegend's Shrinking Hierarchy/@comment-1605058-20170211175133

I'm not necessarily saying this is a flaw with your function, but the fractional iteration as you define it doesn't satisfy a property which would be much desired for it, namely \(f^x(f^y(z))=f^{x+y}(z)\). For consider, for example, \(f_1^{1/2}(n)=f_0^{n/2}(n)=\frac{n^2}{4}\), so \(f_1^{1/2}(f_1^{1/2}(n))=\frac{(n^2/4)^2}{4}=\frac{n^4}{64}\), while \(f_1^{1/2+1/2}(n)=f_1(n)=f_0^n(n)=\frac{n^2}{2}\) (if I've not made any mistake).