The Sphere Packing Problem in Dimension 7

We present a Fourier-analytic treatment of the sphere packing problem in dimension $7$ centred on the scaled root lattice $E_7/\sqrt{2}$. The argument constructs a radial Schwartz function from a weakly holomorphic modular form of weight $7/2$ for $\Gamma_0(2)$ and verifies the Cohn--Elkies sign conditions through the Hauptmodul value $h_2((1+i)/2)=-64$, the Atkin--Lehner eigenvalue of $\theta_{E_7}$, and positivity of the Fourier coefficients of the auxiliary form. These ingredients give the density bound $\Delta_7\le \pi^3/105$, while $E_7/\sqrt{2}$ attains equality.