On the degree-$2$ Siegel theta series of extremal even unimodular lattices of ranks $48$, $72$, $96$, and $120$

We study degree-$2$ Siegel theta series of extremal even unimodular lattices from the genus-$2$ viewpoint initiated by Ozeki. Using Igusa’s structure theorem, we define a depth filtration on genus-$2$ cusp forms, measured by the total degree in ${\chi }_{10}$ and ${\chi }_{12}$, and relate it to the vanishing of low Fourier–Jacobi coefficients forced by extremality. In ranks $48$, $72$, and $96$, this interaction closes exactly and yields a direct genus-$2$ proof that the degree-$2$ theta series is uniquely determined by extremality (conditional on existence in rank $96$). In rank $120$ (again conditional on existence), the same argument leaves a one-dimensional residual line spanned by ${\chi }_{10}^6$: with ${\chi }_{10}$ in the standard integral normalization, any two such degree-$2$ theta series differ by an integer multiple of ${\chi }_{10}^6$.