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arxiv: 2606.05105 · v2 · pith:NS3BU7CHnew · submitted 2026-06-03 · 🧮 math.MG · math.NT· math.PR

Stochastically evolving ellipsoids with symmetries

Elisha B. Abuya , Nihar Gargava , Yufei Zhao This is my paper

Pith reviewed 2026-06-28 02:38 UTC · model grok-4.3

classification 🧮 math.MG math.NTmath.PR
keywords sphere packinglattice packingdensity boundhigh dimensionellipsoid evolutionsymmetriesstochastic process
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The pith

Lattice sphere packings achieve density at least c N² log log N times 2 to the minus N along an infinite sequence of dimensions.

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

The paper establishes that a stochastic process for evolving ellipsoids, when restricted by additional symmetries, yields lattice sphere packings whose density exceeds prior lower bounds by a log log N factor. This construction produces the improved bound along an infinite sequence of dimensions N rather than for all N. A sympathetic reader cares because sphere packing density governs the efficiency of arranging equal balls without overlap in high-dimensional space, with direct ties to coding and geometry questions. The result follows directly from running the evolution while maintaining the symmetries so that the output remains a lattice at every step.

Core claim

There exists a universal constant c greater than zero such that, along an infinite sequence of dimensions N, lattice sphere packings in R^N exist with density at least c N² log log N 2^{-N}.

What carries the argument

Stochastically evolving ellipsoids with symmetries, which generates the packings by evolving shapes while preserving lattice structure and extracting the density improvement.

If this is right

  • The lower bound on lattice packing density improves by a multiplicative log log N factor in infinitely many dimensions.
  • The construction produces a concrete infinite family of lattices realizing the bound.
  • The ellipsoid evolution remains valid when the symmetries are enforced at each step.

Where Pith is reading between the lines

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

  • The same symmetry restriction might be combinable with other evolution rules to test for further density gains.
  • The result leaves open whether the log log N term can be replaced by a slower-growing but still unbounded function.
  • Numerical tracking of the process in moderate dimensions could reveal how the log log term emerges in practice.

Load-bearing premise

The symmetries can be integrated into the ellipsoid evolution process so that the output lattices attain the extra log log N factor in density.

What would settle it

An explicit computation for some large N in the claimed sequence that shows the constructed packing has density strictly less than c N² log log N 2^{-N} for every fixed c.

read the original abstract

We prove that there is a universal constant $c > 0$ such that, along an infinite sequence of dimensions $N$, there are lattice sphere packings in $\mathbb{R}^N$ of density at least $c N^2 \log\log N \, 2^{-N}$, improving the previous best bound due to Klartag by a $\log\log N$ factor. The proof follows Klartag's stochastic ellipsoid evolution process, subject to the cyclotomic symmetries introduced by Venkatesh.

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

1 major / 0 minor

Summary. The paper proves that there exists a universal constant c > 0 such that, for an infinite sequence of dimensions N, there exist lattice sphere packings in R^N with density at least c N² log log N 2^{-N}. This improves Klartag's prior bound by a log log N factor. The argument adapts Klartag's stochastic ellipsoid evolution process by imposing the cyclotomic symmetry constraints introduced by Venkatesh.

Significance. If correct, the result supplies a concrete logarithmic improvement to the best known constructive lower bounds on high-dimensional lattice packing densities. The approach of restricting the stochastic process to a symmetry-invariant subspace while retaining the quantitative gain is of interest, though the manuscript does not mention machine-checked proofs, reproducible code, or parameter-free derivations.

major comments (1)
  1. [Proof of Theorem 1.1 (or equivalent main theorem section)] The central claim requires that the log log N improvement survives the restriction to the cyclotomic-invariant subspace. The manuscript must explicitly re-derive the tail bounds or variance estimates controlling the probability that a random lattice point lies inside the evolving body (the source of Klartag's gain) under the reduced invariant measure; without these calculations the extra factor is not guaranteed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and the suggestion to strengthen the presentation of the main argument. We address the single major comment below and will incorporate the requested material in a revised version of the manuscript.

read point-by-point responses

  1. Referee: [Proof of Theorem 1.1 (or equivalent main theorem section)] The central claim requires that the log log N improvement survives the restriction to the cyclotomic-invariant subspace. The manuscript must explicitly re-derive the tail bounds or variance estimates controlling the probability that a random lattice point lies inside the evolving body (the source of Klartag's gain) under the reduced invariant measure; without these calculations the extra factor is not guaranteed.

    Authors: We agree that an explicit verification is needed to confirm that the logarithmic improvement persists under the cyclotomic symmetry constraints. In the revised manuscript we will insert a new subsection immediately following the statement of the main theorem that re-derives the relevant tail bounds and variance estimates. The argument proceeds by restricting the underlying Gaussian measure to the invariant subspace fixed by the cyclotomic action; because this subspace is spanned by an orthogonal basis of characters, the coordinate-wise independence properties used by Klartag are preserved up to a uniform multiplicative factor depending only on the degree of the cyclotomic extension. Consequently the same concentration inequalities apply, and the extra log log N factor is recovered with a (slightly smaller) universal constant c. We will also record the precise dependence of all constants on the symmetry group so that the derivation is self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation combines independent prior techniques

full rationale

The paper's central claim is obtained by integrating Klartag's stochastic ellipsoid evolution process with Venkatesh's cyclotomic symmetries. Both source techniques originate from distinct prior authors and are treated as external inputs. No equations or steps in the provided description reduce a prediction to a fitted parameter by construction, invoke a self-citation as the sole justification for a uniqueness claim, or rename a known result under new coordinates. The derivation is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim depends on the successful adaptation of existing stochastic and symmetry techniques from the literature.

axioms (2)
  • domain assumption The stochastic ellipsoid evolution process behaves as described by Klartag in high dimensions
    The proof follows this process.
  • domain assumption Cyclotomic symmetries can be imposed on the ellipsoids without disrupting the density estimates
    Introduced by Venkatesh and used here.

pith-pipeline@v0.9.1-grok · 5608 in / 1235 out tokens · 55132 ms · 2026-06-28T02:38:45.302336+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

40 extracted references · 8 canonical work pages · 1 internal anchor

  1. [1]

    Remarks on the disproof of the unit distance conjecture

    Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang, and Melanie Matchett Wood,Remarks on the disproof of the unit distance conjecture, arXiv:2605.20695

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    Keith Ball,A lower bound for the optimal density of lattice packings, Internat. Math. Res. Notices 1992 (1992), 217–221

    1992

  3. [3]

    Joppe Bos, L´ eo Ducas, Eike Kiltz, Tancr` ede Lepoint, Vadim Lyubashevsky, John M. Schanck, Peter Schwabe, Gregor Seiler, and Damien Stehl´ e,CRYSTALS-Kyber: a CCA-secure module- lattice-based KEM, 2018 IEEE European Symposium on Security and Privacy (EuroS&P), 2018, 353–367

    2018

  4. [4]

    H. F. Blichfeldt,The minimum value of quadratic forms, and the closest packing of spheres, Math. Ann. 101 (1929), 605–608

    1929

  5. [5]

    Armand Borel and Harish-Chandra,Arithmetic subgroups of algebraic groups, Ann. of Math. 75 (1962), 485–535

    1962

  6. [6]

    Marcelo Campos, Matthew Jenssen, Marcus Michelen, and Julian Sahasrabudhe,A new lower bound for sphere packing, arXiv:2312.10026. 19

    work page arXiv

  7. [7]

    Henry Cohn and Noam Elkies,New upper bounds on sphere packings. I, Ann. of Math. 157 (2003), 689–714

    2003

  8. [8]

    Miller, Danylo Radchenko, and Maryna Viazovska, The sphere packing problem in dimension24, Ann

    Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska, The sphere packing problem in dimension24, Ann. of Math. 185 (2017), 1017–1033

    2017

  9. [9]

    Henry Cohn and Abhinav Kumar,Optimality and uniqueness of the Leech lattice among lat- tices, Ann. of Math. 170 (2009), 1003–1050

    2009

  10. [10]

    Henry Cohn and Yufei Zhao,Sphere packing bounds via spherical codes, Duke Math. J. 163 (2014), 1965–2002

    2014

  11. [11]

    Davenport and C

    H. Davenport and C. A. Rogers,Hlawka’s theorem in the geometry of numbers, Duke Math. J. 14 (1947), 367–375

    1947

  12. [12]

    Sci., Springer, 2025

    L´ eo Ducas, Lynn Engelberts, and Paola de Perthuis,Predicting module-lattice reduction, Ad- vances in Cryptology — ASIACRYPT 2025, Lecture Notes in Comput. Sci., Springer, 2025

    2025

  13. [13]

    Cryptogr

    L´ eo Ducas, Eike Kiltz, Tancr` ede Lepoint, Vadim Lyubashevsky, Peter Schwabe, Gregor Seiler, and Damien Stehl´ e,CRYSTALS-Dilithium: a lattice-based digital signature scheme, IACR Trans. Cryptogr. Hardw. Embed. Syst. 2018 (2018), no. 1, 238–268

    2018

  14. [14]

    Roni Edwin,Fourier inequalities and sign uncertainty, arXiv:2505.15994

    work page arXiv

  15. [15]

    Fejes T´ oth,¨Uber einen geometrischen Satz, Math

    L. Fejes T´ oth,¨Uber einen geometrischen Satz, Math. Z. 46 (1940), 83–85

    1940

  16. [16]

    Nihar Prakash Gargava,Lattice packings through division algebras, Math. Z. 303 (2023), Paper No. 18

    2023

  17. [17]

    Nihar Gargava, Vlad Serban, and Maryna Viazovska,Moments of the number of points in a bounded set for number field lattices, arXiv:2308.15275

    work page arXiv

  18. [18]

    Nihar Gargava, Vlad Serban, Maryna Viazovska, and Ilaria Viglino,Module lattices and their shortest vectors, arXiv:2510.12893

    work page arXiv

  19. [19]

    Nihar Gargava and Maryna Viazovska,Mean value for random ideal lattices, arXiv:2411.14973

    work page arXiv

  20. [20]

    Hales,A proof of the Kepler conjecture, Ann

    Thomas C. Hales,A proof of the Kepler conjecture, Ann. of Math. 162 (2005), 1065–1185

    2005

  21. [21]

    Hlawka,Zur Geometrie der Zahlen, Math

    E. Hlawka,Zur Geometrie der Zahlen, Math. Z. 49 (1943), 285–312

    1943

  22. [22]

    G. A. Kabatiansky and V. I. Levenshtein,Bounds for packings on the sphere and in space, Problemy Peredachi Informatsii 14 (1978), 3–25; English transl., Problems Inform. Transmis- sion 14 (1978), 1–17

    1978

  23. [23]

    Koen de Boer, Aurel Page, Radu Toma, and Benjamin Wesolowski,Average hardness of SIVP for module lattices of fixed rank, arXiv:2511.13659

    work page arXiv

  24. [24]

    Seungki Kim,Adelic Rogers integral formula, J. Lond. Math. Soc. (2) 109 (2024), Paper No. e12830, 48 pp

    2024

  25. [25]

    Boaz Klartag,Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid, Invent. Math. (2026). 20

    2026

  26. [26]

    Michael Krivelevich, Simon Litsyn, and Alexander Vardy,A lower bound on the density of sphere packings via graph theory, Internat. Math. Res. Notices 2004 (2004), 2271–2279

    2004

  27. [27]

    V. I. Levenshtein,On bounds for packings inn-dimensional Euclidean space, Dokl. Akad. Nauk SSSR 245 (1979), 1299–1303

    1979

  28. [28]

    Department of Commerce, August 2024

    National Institute of Standards and Technology,Module-lattice-based key-encapsulation mech- anism standard, FIPS Publication 203, U.S. Department of Commerce, August 2024

    2024

  29. [29]

    Department of Commerce, August 2024

    National Institute of Standards and Technology,Module-lattice-based digital signature stan- dard, FIPS Publication 204, U.S. Department of Commerce, August 2024

    2024

  30. [30]

    OpenAI,An OpenAI model has disproved a central conjecture in discrete geometry, 20 May 2026, openai.com/index/model-disproves-discrete-geometry-conjecture/

    2026

  31. [31]

    R. A. Rankin,On the closest packing of spheres inndimensions, Ann. of Math. 48 (1947), 1062–1081

    1947

  32. [32]

    C. A. Rogers,Existence theorems in the geometry of numbers, Ann. of Math. 48 (1947), 994– 1002

    1947

  33. [33]

    C. A. Rogers,The packing of equal spheres, Proc. London Math. Soc. (3) 8 (1958), 609–620

    1958

  34. [34]

    Naser Talebizadeh Sardari and Masoud Zargar,New upper bounds for spherical codes and packings, Math. Ann. 389 (2024), 3653–3703

    2024

  35. [35]

    Schmidt,The measure of the set of admissible lattices, Proc

    Wolfgang M. Schmidt,The measure of the set of admissible lattices, Proc. Amer. Math. Soc. 9 (1958), 390–403

    1958

  36. [36]

    Torquato and F

    S. Torquato and F. H. Stillinger,New conjectural lower bounds on the optimal density of sphere packings, Experiment. Math. 15 (2006), 307–331

    2006

  37. [37]

    Stephanie Vance,Improved sphere packing lower bounds from Hurwitz lattices, Adv. Math. 227 (2011), 2144–2156

    2011

  38. [38]

    Akshay Venkatesh,A note on sphere packings in high dimension, Int. Math. Res. Not. IMRN 2013 (2013), 1628–1642

    2013

  39. [39]

    Viazovska,The sphere packing problem in dimension8, Ann

    Maryna S. Viazovska,The sphere packing problem in dimension8, Ann. of Math. 185 (2017), 991–1015

    2017

  40. [40]

    Elisha B

    Masoud Zargar,Stiefel manifolds and upper bounds for spherical codes and packings, arXiv:2407.10697. Elisha B. Abuya Tel Aviv University, Israel. e-mail:ebabuya@tauex.tau.ac.il Nihar Gargava Universit´ e Paris-Saclay, France. e-mail:nihar.gargava@universite-paris-saclay.fr Yufei Zhao Massachusetts Institute of Technology, USA. e-mail:yufeiz@mit.edu 21

    work page arXiv