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

Regev and Stephens-Davidowitz conjectured that the Gaussian mass $\Theta_\Lambda(t) = \sum_{x \in \Lambda} e^{-t\lVert x\rVert^2}$ of any integral lattice $\Lambda \subset \mathbb{R}^n$ is bounded above by $\Theta_{\mathbb{Z}^n}(t)$. 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 $\Theta_{\mathbb{Z}^8}(t) - \Theta_{E_8}(t) = \theta_2(it/\pi)^4\,\theta_4(it/\pi)^4 > 0$. 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 $\Lambda \mapsto h_\Lambda$; near-sharp sequences are similarly excluded under a uniform summability hypothesis.