Scalar Poisson-Summation Certificates

• Scalar Poisson-summation certificates are scalar Fourier-analytic constructions that use a single even Schwartz function to certify Poisson-type summation formulas in lattice settings.
• They enforce integral-shell saturation by forcing the certificate to match the Gaussian at every nonzero point with integer squared norm, leading to rigidity in dimensions n ≥ 4.
• The method encounters a no-go theorem in dimensions n ≥ 8 due to the strict theta-series gap, prompting exploration of higher-order and equivariant techniques.

Scalar Poisson-summation certificates are scalar Fourier-analytic constructions in which a single function, measure, or scalar identity is used to certify a Poisson-type summation formula or an inequality derived from Poisson summation. In the setting of the Regev–Stephens-Davidowitz Gaussian mass maximality conjecture, the certificate is a single even Schwartz function $h:\mathbb{R}^n\to\mathbb{R}$ designed to prove

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$

simultaneously for all lattices in a prescribed class. The central result is a rigidity phenomenon: in dimensions $n\ge 4$, any scalar certificate sharp at $\mathbb{Z}^n$ must saturate every integer shell, and in dimensions $n\ge 8$ this saturation is incompatible with the strict theta-series gap between $\mathbb{Z}^8$ and $E_8$, so no sharp scalar certificate exists (Kominers, 26 May 2026).

1. Lattice setting and certificate definition

A full-rank lattice $\Lambda\subset \mathbb{R}^n$ is called integral if $\langle x,y\rangle\in\mathbb{Z}$ for all $x,y\in\Lambda$, and unimodular if $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$0. Its dual lattice is

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$1

If $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$2 is integral and unimodular, then $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$3 setwise. The conjecture of Regev and Stephens-Davidowitz asserts that for any integral lattice $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$4 and all $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$5,

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$6

The same source emphasizes that this inequality is false shell-by-shell; in dimension $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$7, for example, $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$8 has more short vectors than $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$9. Any proof must therefore use cancellations across shells rather than termwise comparison (Kominers, 26 May 2026).

The scalar Poisson-summation strategy fixes $n\ge 4$0 and chooses a single even Schwartz function $n\ge 4$1 subject to two inequalities. The first is primal majorization,

$n\ge 4$2

The second is dual nonpositivity,

$n\ge 4$3

where the Fourier transform is normalized by

$n\ge 4$4

Poisson summation,

$n\ge 4$5

then yields, for unimodular integral $n\ge 4$6,

$n\ge 4$7

The certificate is called sharp at $n\ge 4$8 if

$n\ge 4$9

Sharpness is an extremality condition: because $\mathbb{Z}^n$0 for every $\mathbb{Z}^n$1, equality must propagate to all rotated copies of $\mathbb{Z}^n$2.

2. Integral-shell saturation

The rigidity theorem is formulated on the integer squared norm locus

$\mathbb{Z}^n$3

For $\mathbb{Z}^n$4, Lagrange’s four-square theorem implies that every positive integer occurs as the squared norm of some $\mathbb{Z}^n$5. Combined with $\mathbb{Z}^n$6-invariance, the union of the nonzero point sets of all rotations $\mathbb{Z}^n$7 is exactly $\mathbb{Z}^n$8 (Kominers, 26 May 2026).

The saturation theorem states that if $\mathbb{Z}^n$9, $n\ge 8$0, and $n\ge 8$1 is an even Schwartz function satisfying primal majorization and dual nonpositivity on every unimodular integral lattice, then sharpness at $n\ge 8$2 forces

$n\ge 8$3

In the terminology of the paper, any sharp scalar certificate exhibits integral-shell saturation.

The mechanism is termwise rigidity. One applies the certificate inequalities and Poisson summation on each self-dual lattice $n\ge 8$4. Since sharpness at $n\ge 8$5 implies sharpness at $n\ge 8$6, the entire chain of inequalities collapses to equality. The relevant sums are absolutely convergent and consist of terms of fixed sign, so equality of sums forces equality term by term: $n\ge 8$7 Rotation and four-squares then propagate this to all of $n\ge 8$8. The significance is that sharpness at one lattice does not remain local: it rigidifies the certificate on every nonzero shell of integer squared radius.

3. The $n\ge 8$9 obstruction and the no-go theorem

The obstruction in dimensions $\mathbb{Z}^8$0 comes from the strict theta-series gap between $\mathbb{Z}^8$1 and $\mathbb{Z}^8$2. Writing

$\mathbb{Z}^8$3

and using the Jacobi theta nullwerte,

$\mathbb{Z}^8$4

one has

$\mathbb{Z}^8$5

The positivity follows from the series expansions for $\mathbb{Z}^8$6 and the product formula

$\mathbb{Z}^8$7

For orthogonal direct sums,

$\mathbb{Z}^8$8

hence, for $\mathbb{Z}^8$9,

$E_8$0

(Kominers, 26 May 2026).

This gap is incompatible with integral-shell saturation. If a sharp scalar certificate existed in dimension $E_8$1, then by saturation it would interpolate the Gaussian and have vanishing Fourier transform at every nonzero point of integer squared norm. The lattice

$E_8$2

is unimodular integral and self-dual, with integer shell structure. Poisson summation on $E_8$3 would then force

$E_8$4

contradicting the strict inequality above. The resulting no-go theorem states that for $E_8$5 and $E_8$6, every even Schwartz $E_8$7 satisfying the scalar certificate conditions must obey

$E_8$8

Thus the sharp $E_8$9 Gaussian mass bound is unattainable by scalar Poisson certificates in dimensions $\Lambda\subset \mathbb{R}^n$0.

4. Stable lattices, graded families, and near-sharp schemes

The obstruction is not confined to unimodular integral lattices in the narrow sense. A full-rank lattice $\Lambda\subset \mathbb{R}^n$1 is called stable if $\Lambda\subset \mathbb{R}^n$2 and every nonzero sublattice $\Lambda\subset \mathbb{R}^n$3 has covolume at least $\Lambda\subset \mathbb{R}^n$4 in its span, equivalently

$\Lambda\subset \mathbb{R}^n$5

for any basis $\Lambda\subset \mathbb{R}^n$6 of $\Lambda\subset \mathbb{R}^n$7. Unimodular integral lattices are stable. In the stable setting one requires

$\Lambda\subset \mathbb{R}^n$8

for every stable $\Lambda\subset \mathbb{R}^n$9. Because $\langle x,y\rangle\in\mathbb{Z}$0 and $\langle x,y\rangle\in\mathbb{Z}$1 are stable and self-dual, the saturation and no-go arguments apply verbatim (Kominers, 26 May 2026).

The same paper extends the obstruction to orbit-constant graded families. Such a family assigns to each stable lattice $\langle x,y\rangle\in\mathbb{Z}$2 an even Schwartz function $\langle x,y\rangle\in\mathbb{Z}$3 satisfying orbit-constancy,

$\langle x,y\rangle\in\mathbb{Z}$4

together with primal majorization and dual nonpositivity on $\langle x,y\rangle\in\mathbb{Z}$5 and $\langle x,y\rangle\in\mathbb{Z}$6. If

$\langle x,y\rangle\in\mathbb{Z}$7

then for $\langle x,y\rangle\in\mathbb{Z}$8,

$\langle x,y\rangle\in\mathbb{Z}$9

Hence no orbit-constant graded scalar family can both be sharp at $x,y\in\Lambda$0 and certify the sharp inequality uniformly over all stable lattices.

A further extension excludes uniformly summable near-sharp sequences. If $x,y\in\Lambda$1 are even Schwartz functions satisfying the scalar certificate inequalities for all unimodular integral lattices, with slack

$x,y\in\Lambda$2

then on every integral shell one has

$x,y\in\Lambda$3

Thus $x,y\in\Lambda$4 and $x,y\in\Lambda$5 for all $x,y\in\Lambda$6 with $x,y\in\Lambda$7. If, moreover, the restrictions to $x,y\in\Lambda$8 are uniformly absolutely summable, dominated convergence plus Poisson summation forces $x,y\in\Lambda$9, again contradicting the $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$00 gap.

Setting	Result	Range
Single ambient $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$01 on unimodular integral lattices	Integral-shell saturation	$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$02
Single ambient $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$03 on unimodular integral lattices	No sharp certificate for the sharp $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$04 bound	$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$05
Stable scalar certificates	Same saturation and no-go	$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$06, $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$07
Orbit-constant graded families	Sharp uniform certification excluded	$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$08
Uniformly summable near-sharp sequences	Compact near-sharp scheme excluded	$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$09

5. Scope, misconceptions, and routes beyond scalarity

Several clarifications are essential. First, the Gaussian itself is not a scalar certificate. Although

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$10

satisfies primal majorization with equality, its Fourier transform is

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$11

so it violates the required dual nonpositivity. The obstruction therefore concerns the scalar certificate method, not the Gaussian test function itself (Kominers, 26 May 2026).

Second, the no-go theorem does not refute the Gaussian mass maximality conjecture. It refutes a particular proof strategy based on a single ambient Schwartz function with pointwise primal and dual sign conditions. The same source explicitly contrasts scalar certificates with higher-order approaches. Scalar certificates observe only single-point data and cannot access pairwise inner products inherent to integrality, $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$12. Semidefinite-programming frameworks such as Bachoc–Vallentin retain angular information via harmonic kernels and positive semidefiniteness, and are presented as a potential route beyond scalar obstructions.

Third, the obstruction has definite boundaries. It applies to orbit-constant families $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$13, but not to the weaker equivariance condition

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$14

A fully equivariant, lattice-dependent scalar program remains outside the saturation mechanism. Likewise, the present rigidity relies on self-duality, so extending comparable arguments to nonunimodular integral lattices, where $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$15 and $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$16, is open. These limitations matter because they isolate precisely which scalar architectures are excluded and which remain formally unruled.

6. Related scalar certificate paradigms

Across adjacent literatures, the phrase “scalar Poisson-summation certificate” denotes several closely related mechanisms rather than a single formal definition. This suggests a common template: scalar data are chosen so that Poisson summation yields either a pointwise identity, a trace identity, or an extremal inequality.

Setting	Scalar datum	Certified statement
LCA quotient $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$17 (Applebaum, 2016)	Continuous density $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$18	PSF and probabilistic trace formula
One-dimensional FS-pairs (Gonçalves, 2023)	Hermite–Biehler data $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$19 or holomorphic $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$20	$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$21
$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$22-spherical eigenmeasures (Alfes et al., 2024)	Scalar modular-type Fourier series	$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$23 and Poisson-type summation
Whittaker/KL index transforms (Ribeiro et al., 2024)	Scalar index kernel $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$24 or $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$25	Explicit PSF in the index variable
Sum-of-squares formulas (Lev et al., 2020)	Odd tempered distribution $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$26 on $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$27	Exact derivative-corrected Poisson-type identity

In Applebaum’s probabilistic setting on a locally compact abelian group $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$28 with discrete subgroup $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$29 and compact quotient $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$30, the scalar datum is the convolution-semigroup density $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$31. Under assumptions (A1) and (A2), one has the pointwise Poisson summation formula

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$32

and at the identity this becomes the probabilistic trace certificate

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$33

The Gaussian on $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$34 satisfies both the PSF and trace certificates; rotationally invariant $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$35-stable semigroups satisfy the trace formula but may fail the PSF; an adelic semistable construction can make both diverge (Applebaum, 2016).

In the one-dimensional classification of Fourier summation formulas, scalar certificates are encoded by almost periodic holomorphic functions and Hermite–Biehler data. A real-antipodal FS-pair $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$36 with $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$37 is classified through a holomorphic symbol $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$38 on the upper half-plane satisfying bounded-type and almost-periodicity conditions, or, in the nonnegative locally finite case, by a Hermite–Biehler entire function $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$39 with $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$40. The resulting certificate identity is

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$41

with $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$42 reconstructed from the phase zeros of $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$43 and weights $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$44 (Gonçalves, 2023).

A different scalar paradigm appears in modular and metaplectic settings. For $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$45-spherical measures,

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$46

the paper on measures and modular forms shows that the theta map $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$47 is an $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$48-equivariant isomorphism between such measures and modular-type Fourier series. A one-term modular transformation law under $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$49 is exactly a scalar eigenmeasure certificate: $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$50 This recovers Poisson-type formulas associated with holomorphic modular forms, eta-products, and higher-dimensional Hilbert modular forms (Alfes et al., 2024).

The index-transform literature provides another explicit scalar model. With the normalization

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$51

the Whittaker-index paper proves Poisson summation identities for $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$52 and, in the Kontorovich–Lebedev case,

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$53

Applying the scalar PSF gives

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$54

together with related Whittaker, Olevskii, and Lommel summation formulas (Ribeiro et al., 2024).

Lev–Reti’s odd-dimensional sum-of-squares formulas convert radial Poisson summation in $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$55 into a scalar tempered distribution

$$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$56

whose Fourier transform is again supported on the same nodes $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$57 plus a derivative at $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$58. For odd $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$59, pairing this identity with an odd Schwartz function yields an exact derivative-corrected Poisson-type summation formula with weights $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$60 (Lev et al., 2020).

Two further comparison points sharpen the boundary of the scalar paradigm. Nori’s algebraic theory of $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$61- and $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$62-summability defines intrinsic scalar lattice sums without analytic continuation and then certifies them by a regularized tempered-distribution Poisson formula; the algebraic $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$63-sum equals the regularized value $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$64 (Nori, 2012). By contrast, the Braverman–Kazhdan and Schubert-variety literatures retain a certificate viewpoint but leave the scalar regime: the certifying objects are Schwartz spaces on nonabelian or quasi-affine spaces, normalized intertwining operators, and Eisenstein-residue boundary terms rather than a single ambient scalar function (Getz et al., 2017, Choie et al., 2021).

Taken together, these developments show that scalar Poisson-summation certificates form a broad but sharply stratified category. In some settings they yield exact and highly structured identities; in the Gaussian mass maximality problem they instead produce a rigidity theorem and, beyond dimension $$\Theta_\Lambda(t)\le \Theta_{\mathbb{Z}^n}(t), \qquad \Theta_\Lambda(t)=\sum_{x\in \Lambda} e^{-t\|x\|^2},$$65, a definitive no-go result for the sharp scalar strategy.