Complete asymptotic type-token relationship for growing complex systems with inverse power-law count rankings

ArXi:2511.02069v2 Announce Type: replace-cross The growth dynamics of complex systems often exhibit statistical regularities involving power-law relationships. For real finite complex systems formed by countable tokens (animals, words) as instances of distinct types (species, dictionary entries), an inverse power-law scaling $S \sim r^{-\alpha}$ between type count $S$ and type rank $r$, widely known as Zipf's law, is widely observed to varying degrees of fidelity.