Kissing number in dimension 31
Lower bound: the records, digit by digit
- Oct 10, 2026241,077+1%by Kevin Li
- Oct 5, 2026238,666+0.0017%by Rustem Takhanov and Stanislav Yun
- Sep 17, 2026238,662+0.131%by Alexander Kravatskiy
- Feb 11, 2026238,350+0.114%by Chengdong Ma and 8 others
- Nov 17, 2025238,078+2.2%by Chengdong Ma and 8 others
- 2017232,874first recordby Kenz Kallal and 2 others
Coloured and underlined: the digits that changed since the previous record. No chart: since the first published record, neither bound has moved by more than 5% of the gap between them.
Cite this frontier
DiscoverIA. Kissing number in dimension 31 [Frontier, checked on Oct 12, 2026]. DiscoverIA registry. https://discoveria.ai/en/frontieres/kissing-d31/
The exact value lies somewhere in between, within a factor of ×9.18. The lower bound has moved 5 times since 2017, most recently on Oct 10, 2026. The upper bound has not moved since 2024.
The maximum number of unit spheres with disjoint interiors that can touch a central unit sphere in R^31. A valid configuration of N spheres gives the lower bound N; a linear or semidefinite programming bound gives an upper bound.
Exact value unknown: gap 241077 versus 2213487 (ratio 9.182 on Cohn's table consulted 2026-10-13). Lower bounds are explicit constructions (the recent ones announce an exact-arithmetic program check); no named third-party review appears in the sources read. The sources read declare AI use for these lower bounds: PackingStar (238078, 2025-11-17); PackingStar (238350, 2026-02-11); Claude Code (238662, 2026-09-17); Claude Code (Claude Opus 5.5), GPT-6 Astra (238666, 2026-10-05); GPT-6-Astra Pro, Claude Opus 5.5 (241077, 2026-10-10). The latest (241077, Kevin Li, 2026-10-10) has AI use declared by its source. The upper bound (2213487) is a semidefinite programming bound (Cohn's table, credited to Leijenhorst-de Laat); no source read attributes it to an AI.
Every step, as a table : Lower bound (7 points)
Best known configuration (constructions), from the Leech, Kallal-Kan-Wang or Cohn-Li records to the AI-assisted constructions of 2025-2026.
| Date | Exact value | Nature | Authors | Source |
|---|---|---|---|---|
| Oct 10, 2026 | 241077 Current record | published | Kevin Li | github.com |
| Oct 5, 2026 | 238666 Current record | published | Rustem Takhanov, Stanislav Yun | github.com |
| Sep 23, 2026 | 238354 | published | Rustem Takhanov, Stanislav Yun | arxiv.org |
| Sep 17, 2026 | 238662 Current record | published | Alexander Kravatskiy | github.com |
| Feb 11, 2026 | 238350 Current record | published | Chengdong Ma, Théo Tao Zhaowei, Pengyu Li, Minghao Liu, Haojun Chen, Zihao Mao, Yuan Cheng, Yuan Qi, Yaodong Yang | arxiv.org |
| Nov 17, 2025 | 238078 Current record | published | Chengdong Ma, Théo Tao Zhaowei, Pengyu Li, Minghao Liu, Haojun Chen, Zihao Mao, Yuan Cheng, Yuan Qi, Yaodong Yang | arxiv.org |
| 2017 | 232874 Current record | published | Kenz Kallal, Tomoka Kan, Eric Wang | doi.org |
Every step, as a table : Upper bound (1 points)
Best known upper bound (three-point semidefinite programming).
| Date | Exact value | Nature | Authors | Source |
|---|---|---|---|---|
| 2024 | 2213487 Current record | published | David de Laat, Nando Leijenhorst | cohn.mit.edu |
Kissing number (21)
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