Kissing number in dimension 26
Lower bound: the records, digit by digit
- Oct 5, 2026199,868+0.031%by Rustem Takhanov and Stanislav Yun
- Sep 19, 2026199,806+0.633%by Alexander Kravatskiy
- Nov 17, 2025198,550+0.019%by Chengdong Ma and 8 others
- 2017198,512first 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.
What is a bound?Cite this frontier
DiscoverIA. Kissing number in dimension 26 [Frontier, checked on Oct 12, 2026]. DiscoverIA registry. https://discoveria.ai/frontiers/kissing-d26
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The exact value lies somewhere in between, within a factor of ×1.84. The lower bound has moved 3 times since 2017, most recently on Oct 5, 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^26. 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 199868 versus 367775 (ratio 1.841 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 (198550, 2025-11-17); Claude Code (199806, 2026-09-19); Claude Code (Claude Opus 5.5), GPT-6 Astra (199868, 2026-10-05). The latest (199868, Rustem Takhanov, Stanislav Yun, 2026-10-05) has AI use declared by its source. The upper bound (367775) is a semidefinite programming bound (Cohn's table, credited to Leijenhorst-de Laat); no source read attributes it to an AI.
Analogy
Take a marble and surround it with identical marbles that all touch it without overlapping: how many can you fit? Flat on a table, with coins around a coin, a few tries give the answer. In space, with marbles, there is room for them to shift, and it took centuries to prove that no extra one can be squeezed in. The kissing number asks the same question in spaces with more than three dimensions.
Where the picture stops
Beyond three dimensions there are no marbles to move around: a marble becomes a list of numbers, "touching" becomes a distance to compute, and the exact answer is known in only a handful of dimensions.
In the glossary : Kissing number
Every step, as a table : Lower bound (5 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 5, 2026 | 199868 Current record | published | Rustem Takhanov, Stanislav Yun | github.com |
| Sep 23, 2026 | 198552 | published | Rustem Takhanov, Stanislav Yun | arxiv.org |
| Sep 19, 2026 | 199806 Current record | published | Alexander Kravatskiy | github.com |
| Nov 17, 2025 | 198550 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 | 198512 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 | 367775 Current record | published | David de Laat, Nando Leijenhorst | cohn.mit.edu |
Kissing number (21)
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