A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank $32$

The integer lattice ${\mathbb{Z}}^n$ is conjectured to maximize the Gaussian mass ${\mathrm{\Theta }}_L(t)={\sum }_{x\in L}{e}^{-t\mathrm{\parallel }x{\mathrm{\parallel }}^2}$ over the set of stable lattices in ${\mathbb{R}}^n$, for every $t>0$. We prove this sharp inequality for every integral unimodular lattice $L$ of rank $n\le 32$, with equality only at $L\cong {\mathbb{Z}}^n$, and furthermore obtain the strict inequality for every even unimodular lattice of rank $40$. The proof does not use the classification of unimodular lattices in these ranks; rather, it parametrizes integral unimodular theta series as polynomials in the modular function $u={\mathrm{\Delta }}_8\mathrm{/}{\vartheta }_3^8\in (0,1\mathrm{/}64]$, with the few coefficients that arise controlled by norm-$1$ splitting, ADE root counts, and shadow positivity.