Equality in a Reverse Minkowski Shell Bound for Integral Lattices via Spherical Designs

For a full-rank integral lattice $\mathcal{L}\subset {\mathbb{R}}^n$, Regev and Stephens-Davidowitz proved that $$N_{=k}(\mathcal{L}):=\mathrm{\mid }\{y\in \mathcal{L}:\parallel y{\parallel }^2=k\}\mathrm{\mid }\le 2\left(\genfrac{}{}{0px}{}{n+2k-2}{2k-1}\right)\mathrm{.}$$ We classify the equality cases. For $n\ge 2$, equality holds if and only if either $k=1$ and $\mathcal{L}\cong {\mathbb{Z}}^n$, or $n=8$, $k=2$, and $\mathcal{L}\cong E_8$. For $n=1$, equality holds exactly when $\mathcal{L}$ represents $k$. The proof shows that equality is rigid. Saturation of the shell bound forces the normalized norm-$k$ shell to be an antipodal tight spherical $(4k-1)$-design. The associated Delsarte–Goethals–Seidel annihilator polynomial gives an arithmetic root condition, which isolates $E_8$ at $k=2$, rules out $k=3$, and combines with the Bannai–Damerell/Bannai theorem and an elementary circle argument to exclude all remaining cases in dimension at least $2$.