Saturation and No-Go Theorems for Scalar Poisson Certificates of Gaussian Mass Maximality

Regev and Stephens-Davidowitz conjectured that the Gaussian mass ${\mathrm{\Theta }}_{\mathrm{\Lambda }}(t)={\sum }_{x\in \mathrm{\Lambda }}{e}^{-t\parallel x{\parallel }^2}$ of any integral lattice $\mathrm{\Lambda }\subset {\mathbb{R}}^n$ is bounded above by ${\mathrm{\Theta }}_{{\mathbb{Z}}^n}(t)$. For $n\ge 4$, we prove a saturation theorem for the natural scalar Poisson-summation certificates of this conjecture: any such certificate that is sharp at ${\mathbb{Z}}^n$ must interpolate the Gaussian, and have vanishing Fourier transform, at every nonzero point of integer squared norm. Applied to the lattice $E_8\oplus {\mathbb{Z}}^{n-8}$, this rigidity is incompatible with the strict theta-series gap ${\mathrm{\Theta }}_{{\mathbb{Z}}^8}(t)-{\mathrm{\Theta }}_{E_8}(t)={\theta }_2(it\mathrm{/}\pi {)}^4{\theta }_4(it\mathrm{/}\pi {)}^4>0$. Consequently, in dimensions $n\ge 8$, no scalar Poisson certificate can attain the sharp ${\mathbb{Z}}^n$ Gaussian mass bound. The same argument rules out the corresponding scalar certificate strategy for the stable-lattice formulation of the conjecture, and extends to orbit-constant graded families $\mathrm{\Lambda }\mapsto h_{\mathrm{\Lambda }}$; near-sharp sequences are similarly excluded under a uniform summability hypothesis.