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

Abstract: Regev and Stephens-Davidowitz conjectured that the Gaussian mass $ΘΛ(t) = \sum{x \in Λ} e^{-t\lVert x\rVert^2}$ 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 $Λ\mapsto hΛ$; near-sharp sequences are similarly excluded under a uniform summability hypothesis.

• The paper proves that any even Schwartz function achieving the sharp Gaussian mass bound must interpolate the Gaussian at all nonzero integer square lengths while its Fourier transform vanishes at those points.
• It demonstrates that in dimensions n ≥ 8, scalar Poisson certificates fail to establish the upper bound for Gaussian mass, as shown via rigidity arguments applied to lattices like E8 ⊕ Z^(n-8).
• The results extend to stable lattices and orbit-constant certificate families, highlighting intrinsic limitations of scalar verification methods and guiding future work toward higher-order or embedding-sensitive approaches.

Scalar Poisson Certificates and Rigidity in Gaussian Mass Bounds

Overview

The paper "Saturation and No-Go Theorems for Scalar Poisson Certificates of Gaussian Mass Maximality" (2605.26803) conducts a rigorous analysis of the Regev–Stephens-Davidowitz conjecture concerning the maximality of the Gaussian mass $\Theta_\Lambda(t)$ among integral lattices, specifically $\Theta_\Lambda(t) \leq \Theta_{Z^n}(t)$ for all $t > 0$ and $\Lambda \subset \mathbb{R}^n$. The primary focus is on scalar Poisson-summation certificates, the Fourier-analytic paradigm inspired by linear programming bounds, and the conditions under which these certificates could establish the conjecture.

Main Results

Integral-Shell Saturation and Rigidity

The author proves a powerful saturation theorem for scalar Poisson certificates: for $n \geq 4$, any even Schwartz function $h$ yielding a sharp (attained) bound at $Z^n$ must interpolate the Gaussian exactly at every nonzero point of integer squared norm and simultaneously vanish on its Fourier transform at these points. This rigidity is derived from repeated applications of Poisson summation and Lagrange’s four-square theorem, yielding the following dichotomy for sharp certificates:

• $h(x) = e^{-t\|x\|^2}$ and $\hat{h}(x) = 0$ for each $x \neq 0$ with $\Theta_\Lambda(t) \leq \Theta_{Z^n}(t)$0.

In effect, sharpness across all rotations imposes interpolation and annihilation constraints incompatible with any nontrivial theta series differences among integral lattices.

No-Go Consequences in Dimension $\Theta_\Lambda(t) \leq \Theta_{Z^n}(t)$1

Applying the rigidity argument to the lattice $\Theta_\Lambda(t) \leq \Theta_{Z^n}(t)$2, which has a strictly smaller theta series than $\Theta_\Lambda(t) \leq \Theta_{Z^n}(t)$3, shows that scalar Poisson certificates fail to establish the sharp bound in these dimensions. Consequently:

• No such certificate can satisfy $\Theta_\Lambda(t) \leq \Theta_{Z^n}(t)$4 as the upper bound for all unimodular integral lattices in $\Theta_\Lambda(t) \leq \Theta_{Z^n}(t)$5.
• The obstruction is generic and extends to stable lattices, not merely those that are unimodular and integral.

This result is underpinned by the explicit theta function gap:

$\Theta_\Lambda(t) \leq \Theta_{Z^n}(t)$6

for all $\Theta_\Lambda(t) \leq \Theta_{Z^n}(t)$7.

Extensions and Strengthened Obstructions

Three significant extensions broaden the scope of the obstruction:

1. Stable Lattices: The no-go results are valid for the stable-lattice formulation, where $\Theta_\Lambda(t) \leq \Theta_{Z^n}(t)$8 and every sublattice has covolume ≥ 1 in its span.
2. Orbit-Constant Graded Families: The obstruction applies to certificate families $\Theta_\Lambda(t) \leq \Theta_{Z^n}(t)$9 that are $t > 0$0-orbit constant. This includes schemes graded by theta series, modular-form coordinates, or finite lists of $t > 0$1 invariants.
3. Near-Sharp Sequences: Sequences of certificates asymptotically attaining sharp bounds are excluded if their restrictions to $t > 0$2 obey a uniform summability condition.

Thus, any compact limiting approach or canonical parametrization using intrinsic invariants is incompatible with establishing maximal Gaussian mass via scalar certificates.

Technical Context and Numerical Comparisons

The argument is built on classical Poisson summation, Fourier analysis, and modular form identities. The multiplicative structure of theta series for orthogonal direct sums and explicit modular formulas for $t > 0$3 and $t > 0$4 lattices are exploited to solidify the gap. The result does not rely on numerical optimization but on exact destructive interference in interpolation constraints.

Theoretically, this places structural limitations on the Cohn–Elkies-style approach, singling out the coexistence of $t > 0$5 and $t > 0$6 in $t > 0$7 as the key obstruction.

Implications and Future Directions

Practically, the results delimit strategies for proving the Gaussian mass maximality conjecture via scalar certificates and canonical lattice invariants. Theoretically, they establish a class of saturation phenomena for Fourier-analytic bounds, highlighting how sharpness constraints can induce incompatibility with geometric structure.

Several implications for future developments in discrete Fourier analysis and lattice optimization arise:

• Higher-Order and Multi-Point Certificates: Successor frameworks need to incorporate angular or pairwise information (e.g., semidefinite programs using Gram matrices, Bachoc–Vallentin kernels), exploiting integral lattice structure at a finer granularity.
• Equivariant or Embedding-Dependent Families: Certificate families sensitive to embeddings or equivariant under orthogonal transformations are not obstructed by these results, opening potential for non-orbit-constant constructions.
• Nonunimodular Cases: The obstruction is sharp for unimodular settings; nonunimodular integral lattices may require novel analysis given the lack of self-duality in Poisson summation.
• Modular Forms and Secrecy Gain: Modular parametrizations or single-point bounds (e.g., Belfiore–Solé conjecture on secrecy gain) evading global pointwise interpolation are outside the scope of the proven obstruction.

The findings refine the landscape for lattice bounds, forcing exploration outside scalar certificate and orbit-constant regimes if maximality is to be universally established.

Conclusion

This work rigorously demonstrates that sharp scalar Poisson-summation certificates, including orbit-constant canonical graded families and compact limits, cannot establish the Gaussian mass maximality conjecture for integral unimodular lattices in dimension $t > 0$8. The no-go theorems clarify the structural reasons for failure rooted in the theta series gap and the rigidity of interpolation constraints—guiding future analytic and computational research toward higher-order, equivariant, or embedding-sensitive methods.