AI 中文总结
该研究在36维构建科恩 - 埃尔基斯线性规划的显式对偶可行点，证明其球体填充密度界高于已知最佳填充，是32维以上首个此类对偶界，还介绍了精确有理线性规划及相关常数的方法学要点。
AI 中文摘要
我们按照科恩（Cohn）和特里安塔菲洛（Triantafillou）的方法，从\(\Gamma_0(24)\)的权为18的模形式空间构建了36维科恩 - 埃尔基斯（Cohn - Elkies）线性规划的一个显式对偶可行点。该证明表明，36维球体填充密度的两点线性规划界比目前已知的最佳填充——中心密度为\(2^{18}/3^{10}\)的克希尚 - 帕苏帕蒂（Kschischang - Pasupathy）填充——至少高出32.91倍。特别是，没有科恩 - 埃尔基斯辅助函数能证明36维中已知的最佳填充是最优的。据我们所知，这是32维以上任何维度的首个此类对偶界，扩展了科恩 - 特里安塔菲洛（\(d = 12,16,20,28,32\)）、李（\(3 \leq d \leq 13\)）以及德库西 - 爱尔兰 - 多斯特特 - 维亚佐夫斯卡（\(d = 6\)）的表格。该证明是精确的：对偶点是一个有理向量，系数非负性通过精确到\(n = 800\)的精确算术验证，并且通过一个显式的德利涅（Deligne）型尾部界证明了两个相关\(q\)展开式最终的正性，该尾部界的常数通过向外舍入的区间算术进行了验证。两个方法学要点可能具有独立的研究价值：精确有理线性规划的约束生成（割平面）公式，它使尾部安全最优可达；以及德利涅记账常数\(C=\sum_{f,e}|\lambda_{f,e}|\,e^{-(k - 1)/2}\)的锐化、提升感知形式，没有它，36维的有限验证会勉强失败。
英文摘要
We construct an explicit dual-feasible point for the Cohn--Elkies linear program in dimension $36$, built from the space of weight-$18$ modular forms for $Γ_0(24)$ following the method of Cohn and Triantafillou. The certificate shows that the two-point linear programming bound on the sphere packing density in dimension $36$ exceeds the density of the best packing currently known -- the Kschischang--Pasupathy packing, of center density $2^{18}/3^{10}$ -- by a factor of at least $32.91$. In particular, no Cohn--Elkies auxiliary function can certify the best known packing in dimension $36$ as optimal. To our knowledge this is the first such dual bound in any dimension above $32$, extending the table of Cohn--Triantafillou ($d=12,16,20,28,32$), Li ($3\le d\le 13$), and de~Courcy-Ireland--Dostert--Viazovska ($d=6$). The certificate is rigorous and machine-checkable, with exact rational data and certified interval bounds: the dual point is a rational vector, coefficient nonnegativity is verified by exact arithmetic up to $n=800$, and eventual positivity of the two relevant $q$-expansions is proved via an explicit Deligne-type tail bound whose constant is certified with outward-rounded interval arithmetic. Two methodological points may be of independent interest: a constraint-generation (cutting-plane) formulation of the exact rational LP, which is what makes an exact vertex whose Eisenstein data supports the tail argument reachable; and a sharpened, lift-aware form of the Deligne bookkeeping constant that discounts deep oldform lifts, without which the finite verification in dimension $36$ fails (the crossover moves past the verified window).
Comments13 pages. Ancillary files: exact certificate + machine-checkable verifier. v2: substantially revised exposition and verification bundle; main result unchanged
