Exponential rate of the densest sphere packing in ℝ^d: between d² 2^(−d) and 2^(−0.599d)

Exactly solved only in dimensions 1, 2, 3, 8 and 24. Hales proved d = 3 (Annals 2005, with Ferguson), with the Flyspeck formal proof published in 2017; Viazovska proved d = 8 (arXiv 2016) and Cohn, Kumar, Miller, Radchenko and Viazovska d = 24, using the linear programming bound of Cohn and Elkies (2003). The best lower bounds for general packings are (1−o(1)) d log d / 2^(d+1) (Campos, Jenssen, Michelen and Sahasrabudhe, arXiv December 2023), the first asymptotic improvement over Rogers' 1947 bound. Klartag (arXiv April 2025) showed lattice packings of density at least c d² 2^(−d) exist. The best upper bound for 48 years was Kabatiansky–Levenshtein (1978), density at most 2^(−0.5990...d+o(d)); Cohn–Zhao showed the Cohn–Elkies method is at least as strong. On 1 August 2026 OpenAI released a document claiming that the Cohn–Elkies linear program has exact asymptotic strength sqrt(e/(2π)) per dimension, which would give Δ_d ≤ 2^(−(0.6044...+o(1))d), the first exponent improvement since 1978 and a limit for the method; the rate was conjectured in 2020 by Afkhami-Jeddi, Cohn, Hartman, de Laat and Tajdini; the work is unrefereed, and Scientific American (17 September 2026) reported that experts had only partly checked the accompanying Lean files. The gap between exponents about 0.60 and 1 remains. The dimension-8 proof was formally verified in Lean in February 2026 (Hariharan, Viazovska et al., with Math Inc.).

Packing and kissing bounds feed error-correcting codes and the heuristic cost models of lattice sieving attacks. No deployed system depends on the exact asymptotics.

A proof that the exponential rate exists and equals a specific constant, or a verified improvement of the Kabatiansky–Levenshtein exponent by any method. Unrefereed claims do not count until verified.