Absorbing polylogarithmic overhead above the I/O exponent

Let $T$ be a runtime function with $T(n)=O(n^r)$ in the platform's rational-exponent sense, let $q$ be the I/O exponent, and fix natural numbers $d$, $K$ and $C$. Then for every rational $r'>\max(r,q)$, the transfer bound

is again $O(n^{r'})$. In words: a polylogarithmic overhead is absorbed by any positive exponent gap above both the algorithm's exponent and the I/O exponent. This is the asymptotic step that turns the uniform runtime-transfer theorem into a fine-grained threshold equivalence.