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

Abstract

Regev and Stephens-Davidowitz conjectured that the Gaussian mass $Θ_Λ(t) = \sum_{x \in Λ} e^{-t\lVert x\rVert^2}$ of any integral lattice $Λ\subset \mathbb{R}^n$ is bounded above by $Θ_{\mathbb{Z}^n}(t)$. 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 $Θ_{\mathbb{Z}^8}(t) - Θ_{E_8}(t) = θ_2(it/π)^4\,θ_4(it/π)^4 > 0$. 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 $Λ\mapsto h_Λ$; near-sharp sequences are similarly excluded under a uniform summability hypothesis.