AI 中文总结
该研究针对李赫格格，利用24维径向傅里叶插值基构造函数，证明了科恩与库马尔提出的196560辅助函数猜想，相关商的有界性及极限性质也得到验证。
AI 中文摘要
科恩（Cohn）和库马尔（Kumar）在2009年提出猜想：存在径向施瓦茨函数g:ℝ²⁴→ℝ，满足当r≥√6时g(r)≤0，当r≥0时其傅里叶变换ĝ(r)≥0，g(2)>0，且(ĝ(0)-g(0))/g(2)=196560。我们利用24维径向傅里叶插值基构造此类函数：若a₂、b₂分别为半径2处值与径向导数插值的对偶基函数，则g_C=a₂-Cb₂恰好具备李赫格格上泊松求和所需的节点数据；球面填充魔法函数可确定b₂并给出其严格符号。我们证明可去商a₂/b₂与â₂/b₂在所需半直线上有界，非紧性步骤源于插值核中的精确系数提取与S尖点展开，两个商均趋近于(43+240log2)/15，对所有足够大的半径，它们分别位于该极限的两侧，带有显式一阶指数修正项。因此，该仿射族中的容许参数构成非空闭射线，每个成员都能证明猜想等式成立，同一插值基还可恢复所有非平凡李赫格格壳系数。
英文摘要
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.
