A dual linear programming bound for sphere packing in dimension 36

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 ${\mathrm{\Gamma }}_0(24)$ 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 $2^{18}\mathrm{/}{3}^{10}$ – 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).