The 196560 auxiliary-function conjecture for the Leech lattice

Abstract

Cohn and Kumar conjectured in 2009 that there is a radial Schwartz function $g\colon\R^{24}\to\R$ satisfying $g(r)\leq0$ for $r\geq\sqrt6$, $\widehat g(r)\geq0$ for $r\geq0$, $g(2)>0$, and $(\widehat g(0)-g(0))/g(2)=196560$. 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 $g_C=a_2-Cb_2$ 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 $\widehat a_2/\widehat b_2$ 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 $(43+240\log2)/15$; 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.