pith. sign in

arxiv: 2605.26803 · v1 · pith:IDIYGNBKnew · submitted 2026-05-26 · 🧮 math.NT · math.FA· math.MG

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

Scott Duke Kominers This is my paper

Pith reviewed 2026-06-29 16:08 UTC · model grok-4.3

classification 🧮 math.NT math.FAmath.MG
keywords Gaussian massPoisson summationintegral latticestheta seriessaturation theoremE8 latticeno-go theorem
0
0 comments X

The pith

In dimensions n ≥ 8, no scalar Poisson certificate can attain the sharp Z^n Gaussian mass bound.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a saturation theorem for scalar Poisson-summation certificates of the conjecture that the Gaussian mass of any integral lattice is at most that of Z^n. Any such certificate sharp at Z^n for n ≥ 4 must interpolate the Gaussian and have vanishing Fourier transform at every nonzero point of integer squared norm. Imposing this saturation on the lattice E8 ⊕ Z^{n-8} contradicts the known strict positivity of the theta-series difference between Z^8 and E8. As a result, no scalar Poisson certificate achieves the sharp bound in dimensions 8 and higher. The same saturation obstruction rules out the strategy for the stable-lattice version of the conjecture and for certain graded families under summability assumptions.

Core claim

Any scalar Poisson certificate that attains the sharp Gaussian mass bound at Z^n in dimension n ≥ 4 must satisfy the saturation condition of interpolating the Gaussian and vanishing its Fourier transform at every nonzero integer-squared-norm point. For n ≥ 8 this saturation condition applied to E8 ⊕ Z^{n-8} is incompatible with the strict positivity of the theta-series difference Θ_{Z^8}(t) − Θ_{E8}(t). Consequently no scalar Poisson certificate can attain the sharp Z^n Gaussian mass bound in dimensions n ≥ 8.

What carries the argument

The saturation condition on scalar Poisson-summation certificates, requiring interpolation of the Gaussian and vanishing Fourier transform at all nonzero points of integer squared norm.

If this is right

  • No scalar Poisson certificate attains the sharp Gaussian mass bound for n ≥ 8.
  • The same no-go holds for the stable-lattice formulation of the conjecture.
  • The argument extends to orbit-constant graded families Λ ↦ h_Λ.
  • Near-sharp sequences are excluded under a uniform summability hypothesis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Proving the conjecture in high dimensions requires certificate methods beyond the scalar Poisson form.
  • The theta-series gap between Z^8 and E8 functions as a structural obstruction to saturation-based proofs for lattice extremal problems.

Load-bearing premise

The saturation condition applies directly to the lattice E8 ⊕ Z^{n-8} and produces a contradiction with the known strict positivity of Θ_{Z^8}(t) − Θ_{E8}(t).

What would settle it

An explicit scalar Poisson certificate that attains the sharp Z^n bound for some n ≥ 8 without satisfying the full saturation interpolation at E8 ⊕ Z^{n-8} points.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves a saturation theorem stating that any scalar Poisson-summation certificate sharp at the integer lattice Z^n (for n≥4) must interpolate the Gaussian and have vanishing Fourier transform at every nonzero point in R^n with integer squared norm. This rigidity is applied to the unimodular lattice E8 ⊕ Z^{n-8} to derive a contradiction with the independently known strict positivity of Θ_{Z^8}(t) − Θ_{E8}(t) = θ_2(it/π)^4 θ_4(it/π)^4 >0, yielding a no-go result: no such certificate attains the sharp Z^n Gaussian mass bound for n≥8. The argument extends to the stable-lattice formulation and to orbit-constant graded families Λ ↦ h_Λ under a uniform summability hypothesis.

Significance. If the saturation theorem and its extension to all integer-squared-norm points hold, the result supplies a clean obstruction to one natural strategy for the Regev-Stephens-Davidowitz conjecture, narrowing the space of possible proofs in dimensions 8 and higher. The paper correctly imports the external E8 theta gap without circularity and obtains a uniform no-go that applies to both the original and stable-lattice versions; the graded-family extension is a useful additional contribution.

major comments (2)
  1. [Saturation theorem] Saturation theorem (abstract and the derivation of the main rigidity statement): the claim that sharpness at Z^n forces interpolation/vanishing conditions at every nonzero x with ||x||^2 ∈ ℤ>0 (rather than only at the points of Z^n itself) is load-bearing for the subsequent contradiction. The provided abstract does not indicate whether the proof invokes radial symmetry of the certificate, an identity theorem for analytic functions, or another mechanism to extend from the discrete Z^n points to other vectors on the same spheres; without an explicit justification, the conditions do not necessarily apply to the points of E8 ⊕ Z^{n-8}.
  2. [Application to E8 ⊕ Z^{n-8}] Application to E8 ⊕ Z^{n-8} (the paragraph deriving the theta-series contradiction): the argument equates the theta series of E8 and Z^8 via Poisson summation under the saturation conditions. If those conditions are only verified at Z^n points, the Poisson sum for E8 ⊕ Z^{n-8} need not force equality, so the strict positivity of Θ_{Z^8}(t) − Θ_{E8}(t) does not yield an immediate contradiction. This step must be checked against the precise hypotheses of the saturation theorem.
minor comments (2)
  1. The abstract is concise but the introduction would benefit from a one-sentence definition of 'scalar Poisson certificate' before the saturation statement.
  2. Notation for the Fourier transform and the parameter t in the theta series could be cross-referenced to a preliminary section for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and constructive comments. The points raised highlight opportunities to clarify the proof of the saturation theorem. We address each major comment below and will make revisions to improve the exposition.

read point-by-point responses

  1. Referee: [Saturation theorem] Saturation theorem (abstract and the derivation of the main rigidity statement): the claim that sharpness at Z^n forces interpolation/vanishing conditions at every nonzero x with ||x||^2 ∈ ℤ>0 (rather than only at the points of Z^n itself) is load-bearing for the subsequent contradiction. The provided abstract does not indicate whether the proof invokes radial symmetry of the certificate, an identity theorem for analytic functions, or another mechanism to extend from the discrete Z^n points to other vectors on the same spheres; without an explicit justification, the conditions do not necessarily apply to the points of E8 ⊕ Z^{n-8}.

    Authors: The extension from the points of Z^n to all nonzero vectors with integer squared norm relies on the scalar (i.e., radial) nature of the Poisson certificate: the test function and its Fourier transform depend only on the squared Euclidean norm. Since for n ≥ 4 the lattice Z^n contains nonzero vectors realizing every positive integer squared norm, the interpolation and vanishing conditions established at those lattice points extend immediately to all other vectors of the same norm by radial symmetry. This mechanism is detailed in Section 3 of the manuscript (the proof of the saturation theorem). We will revise the abstract to explicitly mention the role of radial symmetry in extending the conditions. revision: yes

  2. Referee: [Application to E8 ⊕ Z^{n-8}] Application to E8 ⊕ Z^{n-8} (the paragraph deriving the theta-series contradiction): the argument equates the theta series of E8 and Z^8 via Poisson summation under the saturation conditions. If those conditions are only verified at Z^n points, the Poisson sum for E8 ⊕ Z^{n-8} need not force equality, so the strict positivity of Θ_{Z^8}(t) − Θ_{E8}(t) does not yield an immediate contradiction. This step must be checked against the precise hypotheses of the saturation theorem.

    Authors: With the saturation conditions now understood to hold at all nonzero points of integer squared norm (via the radial extension explained above), and noting that all vectors in E8 ⊕ Z^{n-8} have integer squared norms, the hypotheses of the saturation theorem apply directly to this lattice. Consequently, the Poisson summation formula under these conditions equates the relevant theta series, leading to the contradiction with the known strict positivity of Θ_{Z^8}(t) − Θ_{E8}(t). The argument in the manuscript already uses the full strength of the saturation theorem as stated; we will add a clarifying remark in the application paragraph to reference the radial extension explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external benchmarks

full rationale

The paper proves its saturation theorem internally via Poisson summation and then invokes the independently established strict positivity Θ_{Z^8}(t) − Θ_{E8}(t) > 0, an external fact from classical theta-series theory. No step reduces a claimed prediction to a fitted input by construction, renames a known result, or relies on a self-citation chain for the load-bearing contradiction. The no-go result for n ≥ 8 is obtained by combining the new rigidity statement with this external gap, satisfying the criteria for a non-circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Poisson summation formula and the known strict theta-series gap for E8, both standard in the field; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Poisson summation formula holds for the lattices and test functions under consideration.
    Defines the scalar certificates themselves.
  • domain assumption The difference Θ_{Z^8}(t) − Θ_{E8}(t) is strictly positive for t > 0.
    Supplies the contradiction once saturation is established.

pith-pipeline@v0.9.1-grok · 5777 in / 1415 out tokens · 44376 ms · 2026-06-29T16:08:22.760899+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Majorization and Gaussian-Mass Maximality for Construction-A Lattices from Binary Self-Dual Codes

    math.NT 2026-06 unverdicted novelty 7.0

    Proves the Regev-Stephens-Davidowitz conjecture on Gaussian mass maximality of the integer lattice for unimodular Construction-A lattices from binary self-dual codes via majorization of half-weight distributions and J...

  2. A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank $32$

    math.NT 2026-05 unverdicted novelty 7.0

    Proves sharp reverse Minkowski inequality for Gaussian mass of integral unimodular lattices in ranks ≤32 (equality only at Z^n) and even unimodular lattices in rank 40 via parametrization of theta series as polynomial...

Reference graph

Works this paper leans on

20 extracted references · 3 canonical work pages · cited by 2 Pith papers · 2 internal anchors

  1. [1]

    Bollauf and Hsuan-Yin Lin, On the maximum flatness factor over unimodular lattices, preprint, arXiv:2403.16932, 2025

    Maiara F. Bollauf and Hsuan-Yin Lin, On the maximum flatness factor over unimodular lattices, arXiv:2403.16932, 2025

    work page arXiv 2025

  2. [2]

    Bollauf, Hsuan-Yin Lin, and yvind Ytrehus, Secrecy gain of formally unimodular lattices from codes over the integers modulo 4 , IEEE Trans

    Maiara F. Bollauf, Hsuan-Yin Lin, and yvind Ytrehus, Secrecy gain of formally unimodular lattices from codes over the integers modulo 4 , IEEE Trans. Inform. Theory 70 (2024), no. 5, 3309--3329

    2024

  3. [3]

    2010 International Symposium on Information Theory and its Applications (ISITA), 2010, pp

    Jean-Claude Belfiore and Fr \'e d \'e rique Oggier, Secrecy gain: a wiretap lattice code design, Proc. 2010 International Symposium on Information Theory and its Applications (ISITA), 2010, pp. 174--178

    2010

  4. [4]

    2010 IEEE Information Theory Workshop (ITW), 2010, pp

    Jean-Claude Belfiore and Patrick Sol \'e , Unimodular lattices for the G aussian wiretap channel , Proc. 2010 IEEE Information Theory Workshop (ITW), 2010, pp. 1--5

    2010

  5. [5]

    Christine Bachoc and Frank Vallentin, New upper bounds for kissing numbers from semidefinite programming, J. Amer. Math. Soc. 21 (2008), no. 3, 909--924

    2008

  6. [6]

    Henry Cohn and Noam Elkies, New upper bounds on sphere packings I , Ann. of Math. 157 (2003), no. 2, 689--714

    2003

  7. [7]

    Henry Cohn and Abhinav Kumar, Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007), no. 1, 99--148

    2007

  8. [8]

    Miller, Danylo Radchenko, and Maryna Viazovska, Universal optimality of the E_8 and L eech lattices and interpolation formulas , Ann

    Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska, Universal optimality of the E_8 and L eech lattices and interpolation formulas , Ann. of Math. 196 (2022), no. 3, 983--1082

    2022

  9. [9]

    Conway and Neil J

    John H. Conway and Neil J. A. Sloane, Sphere Packings, Lattices and Groups , 3rd ed., Grundlehren der Mathematischen Wissenschaften, vol. 290, Springer-Verlag, New York, 1999

    1999

  10. [10]

    Henry Cohn and Nicholas Triantafillou, Dual linear programming bounds for sphere packing via modular forms, Math. Comp. 91 (2022), no. 333, 491--508

    2022

  11. [11]

    Matthew de Courcy-Ireland, Maria Dostert, and Maryna Viazovska, Six-dimensional sphere packing and linear programming, Math. Comp. 93 (2024), no. 348, 1993--2029

    2024

  12. [12]

    Anne-Maria Ernvall-Hyt \"o nen, On a conjecture by B elfiore and S ol \'e on some lattices , IEEE Trans. Inform. Theory 58 (2012), no. 9, 5950--5955

    2012

  13. [13]

    Scott Duke Kominers, Equality in a reverse M inkowski shell bound for integral lattices via spherical designs , arXiv:2605.25126, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  14. [14]

    Rupert Li, Dual linear programming bounds for sphere packing via discrete reductions, Adv. Math. 460 (2024), Art. 110043

    2024

  15. [15]

    2011 IEEE Information Theory Workshop (ITW), 2011, pp

    Fuchun Lin and Fr \'e d \'e rique Oggier, Secrecy gain of G aussian wiretap codes from unimodular lattices , Proc. 2011 IEEE Information Theory Workshop (ITW), 2011, pp. 718--722

    2011

  16. [16]

    Fr \'e d \'e rique Oggier, Patrick Sol \'e , and Jean-Claude Belfiore, Lattice codes for the wiretap G aussian channel: construction and analysis , IEEE Trans. Inform. Theory 62 (2016), no. 10, 5690--5708

    2016

  17. [17]

    Oded Regev and Noah Stephens-Davidowitz, A reverse M inkowski theorem , Ann. of Math. 199 (2024), no. 1, 1--49

    2024

  18. [18]

    Number Theory 279 (2026), 256--266

    , A simple proof of a reverse M inkowski theorem for integral lattices , J. Number Theory 279 (2026), 256--266

    2026

  19. [19]

    Viazovska, The sphere packing problem in dimension 8 , Ann

    Maryna S. Viazovska, The sphere packing problem in dimension 8 , Ann. of Math. 185 (2017), no. 3, 991--1015

    2017

  20. [20]

    Nina Zubrilina, Zeros of optimal functions in the C ohn-- E lkies linear program , arXiv:1906.11112, 2019

    work page internal anchor Pith review Pith/arXiv arXiv 1906