The 196560 auxiliary-function conjecture for the Leech lattice

Abstract

Cohn and Kumar conjectured in 2009 that there is a radial Schwartz function $g:{\mathbb{R}}^{24}\to \mathbb{R}$ satisfying $g(r)\le 0$ for $r\ge \sqrt{6}$, $\hat{g}(r)\ge 0$ for $r\ge 0$, $g(2)>0$, and $(\hat{g}(0)-g(0))/g(2)=196560$. We construct such functions from the radial Fourier interpolation basis in dimension $24$. If $a_2,b_2$ denote the basis functions dual to value and radial-derivative interpolation at radius $2$, then $g_C=a_2-C{b}_2$ has exactly the nodal data needed for Poisson summation over the Leech lattice. The sphere-packing magic function identifies $b_2$ and supplies its strict signs. We prove that the removable quotients $a_2/b_2$ and ${\hat{a}}_2/{\hat{b}}_2$ are bounded on the required half-lines. The noncompact step follows from exact coefficient extraction in the interpolation kernel and an $S$-cusp expansion. Both quotients tend to $(43+240\log 2)/15$; for all sufficiently large radii they lie on opposite sides of this limit, with an explicit first exponential correction. Consequently the admissible parameters in this affine family form a nonempty closed ray, and every member proves the conjectured identity. The same interpolation basis recovers every nontrivial Leech-shell coefficient.