HEXAGONbeta

A Fourier Consequence for Lattice Kissing Numbers and Average Contacts

A mix of human-written and AI-generated textHuman understanding: all partsmath.MG — Metric Geometrymath.CO — Combinatoricsmath.NT — Number Theory

Contributed by Scott Kominers ↗

Version 1 / Oct 02, 2026 / CC BY-SA 4.0

Abstract

We deduce a new asymptotic upper bound Klat(n)≤(2e/π+o(1))nK_{\mathrm{lat}}(n)\leq\left(\sqrt{2e/\pi}+o(1)\right)^n on the maximal lattice kissing number Klat(n)K_{\mathrm{lat}}(n) in dimension nn, using the explicit auxiliary functions constructed in OpenAI's "Ten Advances" preprint. The same bound holds for the average contact degree of a finite packing of congruent balls. The bound's base-22 exponential rate rounds to 0.39560.3956, matching the value extrapolated empirically by Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini in 2020.

Provenance statement

I used LLMs to assist with analysis and synthesis in the preparation of this article, particularly GPT-6 Astra and Claude Fable 5.1 (accessed in part via Poe with the support of Quora, where I am an advisor). In particular, the bound I present was developed in dialogue with GPT-6 Astra. In addition, I greatly appreciate helpful comments from Henry Cohn and Thijs Laarhoven.

Tools used

OpenAI
GPTVersion 6 Astra
Anthropic
Claude FableVersion 5.1
Quora
Poe

References

  1. Nima Afkhami-Jeddi, Henry Cohn, Thomas Hartman, David de Laat, and Amirhossein Tajdini, High-dimensional sphere packing and the modular bootstrap , J. High Energy Phys. 2020 (2020), no. 12, 066, https://doi.org/10.1007/JHEP12(2020)066 doi:10.1007/JHEP12(2020)066 .DOI
  2. Kurt M. Anstreicher, Improved linear programming bounds for antipodal spherical codes , Discrete Comput. Geom. 28 (2002), no. 1, 107--114, https://doi.org/10.1007/s00454-001-0080-5 doi:10.1007/s00454-001-0080-5 .DOI
  3. E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of Abelian groups , J. Austral. Math. Soc. 1 (1959), no. 1, 47--63, https://doi.org/10.1017/S1446788700025064 doi:10.1017/S1446788700025064 .DOI
  4. Huck Bennett, Alexander Golovnev, and Noah Stephens-Davidowitz, Difficulties constructing lattices with exponential kissing number from codes , IEEE Trans. Inform. Theory 71 (2025), no. 10, 7644--7648, https://doi.org/10.1109/TIT.2025.3593195 doi:10.1109/TIT.2025.3593195 .DOI
  5. K á roly Bezdek and Muhammad A. Khan, Contact numbers for sphere packings , New Trends in Intuitive Geometry , Bolyai Soc. Math. Stud., vol. 27, Springer, Berlin, Heidelberg, 2018, https://doi.org/10.1007/978-3-662-57413-3_2 doi:10.1007/978-3-662-57413-3_2 , pp. 25--47.DOI
  6. Henry Cohn and Noam Elkies, New upper bounds on sphere packings I , Ann. of Math. (2) 157 (2003), no. 2, 689--714, https://doi.org/10.4007/annals.2003.157.689 doi:10.4007/annals.2003.157.689 .DOI
  7. Henry Cohn and Yufei Zhao, Sphere packing bounds via spherical codes , Duke Math. J. 163 (2014), no. 10, 1965--2002, https://doi.org/10.1215/00127094-2738857 doi:10.1215/00127094-2738857 .DOI
  8. P. Delsarte, J. M. Goethals, and J. J. Seidel, Spherical codes and designs , Geom. Dedicata 6 (1977), no. 3, 363--388, https://doi.org/10.1007/BF03187604 doi:10.1007/BF03187604 .DOI
  9. Maria Dostert, Alexander Kolpakov, and Fernando M á rio de Oliveira Filho, Semidefinite programming bounds for the average kissing number , Israel J. Math. 247 (2022), no. 2, 635--659, https://doi.org/10.1007/s11856-022-2288-4 doi:10.1007/s11856-022-2288-4 .DOI
  10. Maxime Fortier Bourque and Bram Petri, Kissing numbers of closed hyperbolic manifolds , Amer. J. Math. 144 (2022), no. 4, 1067--1085, https://doi.org/10.1353/ajm.2022.0023 doi:10.1353/ajm.2022.0023 .DOI
Show all 22 referencesShow fewer references
  1. D. V. Gorbachev, Extremal problem for entire functions of exponential spherical type, connected with the Levenshtein bound on the sphere packing density in Rn R ^n , Izv. Tul. Gos. Univ. Ser. Mat. Mekh. Inform. 6 (2000), no. 1, 71--78 (Russian).
  2. G. A. Kabatiansky and V. I. Levenshtein, On bounds for packings on a sphere and in space , Problems Inform. Transmission 14 (1978), no. 1, 1--17.
  3. Thijs Laarhoven and Scott Duke Kominers, A lattice family with kissing numbers (Ln)e2n ( L _n) e^ 2 n , https://arxiv.org/abs/2609.13608v2 arXiv:2609.13608v2 , 2026.arXiv
  4. Gabriele Nebe, The second minimum of Barnes--Wall lattices , https://arxiv.org/abs/2603.23133v2 arXiv:2603.23133v2 , 2026.arXiv
  5. OpenAI , Ten advances in mathematics and theoretical computer science , 2026, technical manuscript, version of August 6, 2026. https://cdn.openai.com/pdf/ten-proofs-oai.pdf (accessed September 26, 2026).link
  6. Alex Samorodnitsky, On linear programming bounds for spherical codes and designs , Discrete Comput. Geom. 31 (2004), no. 3, 385--394, https://doi.org/10.1007/s00454-003-2858-0 doi:10.1007/s00454-003-2858-0 .DOI
  7. Naser Talebizadeh Sardari and Masoud Zargar, New upper bounds for spherical codes and packings , Math. Ann. 389 (2024), 3653--3703, https://doi.org/10.1007/s00208-023-02738-z doi:10.1007/s00208-023-02738-z .DOI
  8. V. M. Sidel'nikov, New bounds for densest packing of spheres in nn -dimensional Euclidean space , Math. USSR-Sb. 24 (1974), no. 1, 147--157, https://doi.org/10.1070/SM1974v024n01ABEH001911 doi:10.1070/SM1974v024n01ABEH001911 .DOI
  9. Nils-Peter Skoruppa, Quick asymptotic upper bounds for lattice kissing numbers , Mathematika 49 (2002), no. 1--2, 51--57, https://doi.org/10.1112/S0025579300016041 doi:10.1112/S0025579300016041 .DOI
  10. Serge Vl a du t , Lattices with exponentially large kissing numbers , Mosc. J. Comb. Number Theory 8 (2019), no. 2, 163--177, retracted in 2025.
  11. G. L. Watson, The number of minimum points of a positive quadratic form , Dissertationes Math. (Rozprawy Mat.), no. 84, Instytut Matematyczny Polskiej Akademii Nauk, Warszawa, 1971.
  12. Masoud Zargar, Stiefel manifolds and upper bounds for spherical codes and packings , https://arxiv.org/abs/2407.10697 arXiv:2407.10697 , 2024.arXiv

Version history

  1. v1Submitted by Scott KominersInitial depositCurrentOct 02, 2026