Cohn–Miller rationality conjecture for optimal radial functions

Let $f$ and $\widehat{f}$ be optimal radial functions for the Euclidean linear programming sphere-packing bound, normalized so that

$$f(0)=\widehat{f}(0)=1.$$

Their quadratic Taylor coefficients are the coefficients of the degree-two terms in their Taylor expansions at the origin. Cohn–Miller conjecture. The quadratic Taylor coefficients of the optimal radial functions $f$ and $\widehat{f}$, normalized as above with $f(0)=\widehat{f}(0)=1$, are rational numbers when $n=8$ or $n=24$, as shown in Table~. The observed rationality reflects additional unexplained structure in the hypothetical auxiliary functions for the sharp sphere-packing bounds in dimensions $8$ and $24$.