Kissing number in dimension 34
- upper bound 6,422,593by David de Laat and Nando Leijenhorstcohn.mit.edu
- lower bound 384,196by Shuxing Yang and 15 othersarxiv.org
- Last checked
Cite this frontier
DiscoverIA. Kissing number in dimension 34 [Frontier, checked on Oct 12, 2026]. DiscoverIA registry. https://discoveria.ai/frontiers/kissing-d34
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The exact value lies somewhere in between, within a factor of ×16.72. The lower bound has not moved since 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^34. 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 384196 versus 6422593 (ratio 16.72 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: Qiushi Engine (384196, 2026-09-28). The latest (384196, Shuxing Yang, Rui Zhao, Junyao Wu et al., 2026-09-28) has AI use declared by its source. The upper bound (6422593) 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 (1 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 |
|---|---|---|---|---|
| Sep 28, 2026 | 384196 Current record | published | Shuxing Yang, Rui Zhao, Junyao Wu, Yize Wang, Fujia Chen, Kaihao Zhu, Wenhao Li, Zichen Li, Yaqi Li, Shenzhan Hong, Yuang Pan, Junjie Yang, Taowen Deng, Jincheng Mi, Hongsheng Chen, Yihao Yang | arxiv.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 | 6422593 Current record | published | David de Laat, Nando Leijenhorst | cohn.mit.edu |
Kissing number (21)
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