Kissing number problem
In a given dimension, how many equal spheres can touch a central sphere of the same size without overlapping?
Summary
AI-assisted claims
Kissing numbers — Preprints and improved kissing number constructions
“We prove that the kissing numbers in 19 and 21 dimensions are at least 12460 and 31560, improving the best published bounds 11948 (Ho) and 29768 (Cohn and Li). … We prove that the kissing numbers in 28, 29, 30 and 31 dimensions are at least 205172, 210495, 222073 and 241079. … We give two kissing configurations of 8358 points in 18 dimensions. This exceeds the best published lower bound, 7654 by Cohn and Li, by 704, and ties a value reported earlier by Kravatskiy in a public repository.”
- By
- Kevin Li (GitHub: Felpix-Studios)
- AI involvement
- GPT-6 Astra Pro, Claude Opus 5.5 and Claude Fable 5.1 were used to find these constructions, prepare supplementary data and draft this paper. The manuscript was proofread and lightly revised by the author.
- Model
- GPT-6 Astra Pro; Claude Opus 5.5; Claude Fable 5.1, as stated
- Where
- GitHub repository Felpix-Studios/kissing-number
- Checked
- 18 Oct 2026
Updates
- 13 Oct 2026 · Revised by Kevin Li: “bound updates”
- 14 Oct 2026 · Revised by Kevin Li: “Pre-print updates”
New lower bounds for kissing numbers in dimensions 25, 26, 27, 29 and 31
“Starting from constructions based on the Leech lattice, we obtain the improved lower bounds τ25 ≥ 197583, τ26 ≥ 199868, τ27 ≥ 201589, τ29 ≥ 209970 and τ31 ≥ 238666.”
- By
- Rustem Takhanov, Stanislav Yun
- AI involvement
- All new constructions were generated with AI systems: Claude Code with Claude Opus 5.5, and GPT-6 Astra.
- Model
- Claude Code with Claude Opus 5.5; GPT-6 Astra, as stated
- Where
- GitHub repository k-nic/kissing_d25-31
- Checked
- 14 Oct 2026
A Fourier Consequence for Lattice Kissing Numbers and Average Contacts
“We deduce a new asymptotic upper bound K_lat(n) ≤ (√(2e/π) + o(1))^n on the maximal lattice kissing number K_lat(n) in dimension n, using the explicit auxiliary functions constructed in OpenAI's "Ten Advances" preprint.”
- By
- Scott Kominers
- AI involvement
- Text declaration: A mix of human-written and AI-generated. Human-understanding declaration: All parts of this contribution have been checked and understood by a human. Additional provenance: 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.
- Model
- GPT-6 Astra, Claude Fable 5.1, as stated
- Where
- Hexagon
- Checked
- 27 Oct 2026
Improved kissing numbers in nineteen and twenty-one dimensions
“We prove that the kissing numbers in 19 and 21 dimensions are at least 12276 and 30779.”
- By
- Kevin Li (GitHub: Felpix-Studios)
- AI involvement
- GPT-6-Astra Pro and Claude Opus 5.5 were used to find these constructions, write the Lean formalization, and draft the paper.
- Model
- GPT-6-Astra Pro; Claude Opus 5.5, as stated
- Where
- GitHub repository Felpix-Studios/D19-D21-Kissing
- Formal proof
- Lean 4, published by the source
- Checked
- 14 Oct 2026
Updates
- 04 Oct 2026 · Revised by Kevin Li: “τ₂₁ ≥ 30779 (best published bound 29768: H. Cohn and A. Li, arXiv:2411.04916).”
- 05 Oct 2026 · Revised by Kevin Li: “D19: tau19 >= 12270 (restore two deleted points)”
- 10 Oct 2026 · Revised by Kevin Li: “τ₁₉ ≥ 12276 (best published bound 11948: B. S. Ho, arXiv:2603.10425),”
Large-Scale Autonomous Discovery of Kissing Number Constructions
“Here we use Qiushi Engine, an autonomous multi-agent research system, to investigate kissing numbers and obtain new lower bounds in nineteen dimensions: 25, 27, 32–39, 43, 45, and 49–55.”
- By
- 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
- AI involvement
- The autonomous system carried out the construction searches, mathematical analysis and computational verification, while each final result was reduced to explicit mathematical arguments and independently checkable finite certificates.
- Model
- Qiushi Engine, as stated
- Where
- arXiv
- Checked
- 14 Oct 2026
Kissing numbers: new lower bounds in dimensions 18 through 96, with verification packages
“Verification packages for new lower bounds on the kissing number in 53 dimensions: 18, 25-31, 38, 39, 49-63, 68-71 and 73-96.”
- By
- Alexey Kravatskiy
- AI involvement
- The constructions, the verification code and the write-ups in this repository were developed by the author working with Claude Code (Anthropic).
- Model
- Claude Code (Anthropic), as stated
- Where
- GitHub repository alexlegeartis/KissingNumbers
- Checked
- 14 Oct 2026
Updates
- 07 Sep 2026 · Revised by Alexey Kravatskiy: “Dimension 25: tau(25) >= 197569, the lens-head configuration, replaces the +2 cap-level claim”
- 15 Sep 2026 · Revised by Alexey Kravatskiy: “Dimensions 28 to 31: four new claims, 52 in all; the norm-8 frame layer is 96k points in every dimension”
- 16 Sep 2026 · Revised by Alexey Kravatskiy: “Debugging pass: two claims that had outgrown their scope, and a check that could not fail”
Ten Advances in Mathematics and Theoretical Computer Science
“The n-dimensional kissing number is τ_n = A(n, 1/2). Already hierarchy level 2 gives τ_n ≤ 2^((0.39661+o(1))n), compared with the optimized classical exponent 0.400944….”
- By
- OpenAI
- AI involvement
- The results were achieved by an internal version of Astra, our next major model. … These arguments were then prepared into manuscripts by humans with the same model. Afterward, the model formalized each argument in a Lean certificate.
- Model
- an internal version of Astra, as stated
- Where
- OpenAI (paper PDF accompanying the post 'Ten advances in mathematics and theoretical computer science')
- Checked
- 14 Oct 2026
Updates
Harnessing the Collective Intelligence of AI Agents in the Wild for New Discoveries
“As of May 2026, agents on EinsteinArena have discovered 12 new state-of-the-art results better than any previous human or AI solutions. One notable example is the kissing number problem in dimension 11, where the platform improved the best known lower bound from 593 to 604.”
- By
- Federico Bianchi, Yongchan Kwon, Aneesh Pappu, James Zou
- AI involvement
- In EinsteinArena, a collaborative effort among AI agents improved this lower bound to 594, after which we had agents build on their result to further extend it to 604.
- Where
- arXiv
- Checked
- 14 Oct 2026
Updates
A new lower bound for the kissing number in 19 dimensions
“We prove that the kissing number in 19 dimensions is at least 11948, improving the bound of Cohn and Li by 256.”
- By
- Boon Suan Ho
- AI involvement
- GPT-5.4 Pro was used in the discovery of the construction and in revising the exposition.
- Model
- GPT-5.4 Pro, as stated
- Where
- arXiv
- Checked
- 14 Oct 2026
Updates
Finding Kissing Numbers with Game-theoretic Reinforcement Learning
“Working within extremal configuration spaces, PackingStar discovers new interpretable geometric structures that improve 15 strong bounds held for decades in kissing numbers and their generalizations, several of them provably optimal under natural inner products.”
- By
- Chengdong Ma, Théo Tao Zhaowei, Pengyu Li, Minghao Liu, Haojun Chen, Zihao Mao, Yuan Cheng, Yuan Qi, Yaodong Yang
- AI involvement
- Accordingly, we recast this problem as a cooperative matrix-completion game, and train a reinforcement learning system, PackingStar, to solve it. One player fills cosine entries while the other corrects suboptimal ones, making explosive geometric complexity tractable.
- Model
- PackingStar (reinforcement learning system), as stated
- Where
- arXiv
- Checked
- 14 Oct 2026
AlphaEvolve: A Gemini-powered coding agent for designing advanced algorithms
“For example, it advanced the kissing number problem. This geometric challenge has fascinated mathematicians for over 300 years and concerns the maximum number of non-overlapping spheres that touch a common unit sphere. AlphaEvolve discovered a configuration of 593 outer spheres and established a new lower bound in 11 dimensions.”
- By
- AlphaEvolve team, Google DeepMind
- AI involvement
- AlphaEvolve leverages an ensemble of state-of-the-art large language models: our fastest and most efficient model, Gemini Flash, maximizes the breadth of ideas explored, while our most powerful model, Gemini Pro, provides critical depth with insightful suggestions.
- Model
- AlphaEvolve (ensemble of Gemini Flash and Gemini Pro), as stated
- Where
- Google DeepMind blog
- Checked
- 14 Oct 2026
Claims as their sources state them. Papex has not checked them; the status stays the source’s.
Why it is open
Each dimension is a separate problem. An arrangement of touching spheres gives a lower bound on τn, and a proof that no larger arrangement exists gives an upper bound; lower bounds usually come from explicit constructions, and upper bounds from linear programming and its extensions [1]. A dimension is settled when the two bounds meet. That has happened in six dimensions: τ1 = 2 and τ2 = 6, which are easy, τ3 = 12, τ4 = 24, τ8 = 240 and τ24 = 196560 [1].
Even three dimensions was hard: twelve spheres at the corners of a regular icosahedron leave gaps, so they can be moved into infinitely many arrangements, which Christine Bachoc and Frank Vallentin give as part of the reason [5]. Peter Boyvalenkov, Stefan Dodunekov and Oleg Musin call dimension five "the first open case" [1]. On 14 October 2026 Henry Cohn's table of the best known bounds gave 40 ≤ τ5 ≤ 44 and 604 ≤ τ11 ≤ 868, and the ratio of upper to lower bound reached about 60 in dimensions 42 and 43 [9].
In high dimensions even the growth rate is unknown: τn lies between about 2^(0.2075n) and 2^(0.401n) [1, 8]. The kissing problem is closely tied to the wider problems of spherical codes, sets of points on a sphere with a minimum separation, and of sphere packing, the densest arrangement of equal balls in space [1]. No prize is attached to it.
What is known
Three and four dimensions (1694–2008). Whether a thirteenth sphere fits around one in space was the subject of a dispute between Isaac Newton and David Gregory in 1694; Newton's answer, 12, was right but unproved, some nineteenth-century proofs were incomplete, and K. Schütte and B. L. van der Waerden gave the first detailed proof in 1953 [1, 2]. In the 1970s P. Delsarte developed a linear-programming method for upper bounds, based on polynomials whose sums over all pairs of points of any configuration are non-negative [5]. In three and four dimensions it gives only τ3 ≤ 13 and τ4 ≤ 25 [5], and V. V. Arestov and A. G. Babenko proved in 2000 that pure linear programming cannot do better in dimension four [1]. A 1993 proof by W.-Y. Hsiang that τ4 = 24 "was not widely recognized as complete" [1]. Musin proved τ4 = 24 in a 2003 preprint, published in 2008, with a modification of Delsarte's method [1, 4].
Eight and twenty-four dimensions (1979). V. I. Levenshtein and, independently, A. M. Odlyzko and N. J. A. Sloane used linear programming to prove τ8 = 240 and τ24 = 196560 [1, 3, 5]. These are the numbers of shortest vectors in the E8 lattice and the Leech lattice, and E. Bannai and Sloane later proved that these optimal arrangements are unique [5].
Semidefinite programming (2008–2024). Bachoc and Vallentin added conditions on triples of points, giving a semidefinite program, now called the three-point bound, that recovers the known values in dimensions 3, 4, 8 and 24 and lowered the upper bounds in dimensions 5, 6, 7, 9 and 10 [5, 6]. Later computations by Hans Mittelmann and Vallentin, and by Fabrício Machado and Fernando de Oliveira Filho, give the current bounds of 44 in dimension 5 and 553 in dimension 10, respectively [9]. In 2024 Nando Leijenhorst and David de Laat computed improved three-point bounds in dimensions 11 to 23, noting that they did not carry out the interval-arithmetic step needed to make them rigorous [6]; Cohn's table uses them [9]. A 2024 preprint by de Laat, Leijenhorst and Willem de Muinck Keizer, not peer reviewed, computes the second level of the Lasserre hierarchy of semidefinite programs, whose first level is the linear-programming bound, and reports a proof that the 24 vertices of the regular 24-cell form the unique optimal arrangement in dimension four, together with τ6 ≤ 77 [7]; in 2008 Bachoc and Vallentin had called uniqueness there "widely believed, but remains unproven" [5].
High dimensions. A configuration to which no further sphere can be added already contains about (2/√3)^n spheres, up to a polynomial factor, and G. A. Kabatiansky and Levenshtein proved the upper bound of about 2^(0.401n) in 1978 [1, 8]. Matthew Jenssen, Felix Joos and Will Perkins raised the lower bound by a factor proportional to n with a random "hard sphere" model, and Irene Gil Fernández, Jaehoon Kim, Hong Liu and Oleg Pikhurko gained a further factor of about 3.44; the bounds remain exponentially far apart [8].
Recent claims. In May 2025 Google DeepMind reported that AlphaEvolve, a coding agent that uses an ensemble of Gemini Flash and Gemini Pro models, found an arrangement of 593 spheres in dimension 11, above the previous record of 592 [10]. In November 2025 Chengdong Ma and colleagues reported that PackingStar, a reinforcement-learning system that treats the search as a cooperative matrix-completion game, improved 15 bounds for kissing numbers and their generalisations, including lower bounds in dimensions 25 to 31 [11]. In March 2026 Boon Suan Ho reported that τ19 ≥ 11948, 256 more than the bound of Cohn and Anqi Li; the paper says that "GPT-5.4 Pro was used in the discovery of the construction and in revising the exposition", and a second version on 20 March corrected errors with the main results unchanged [12]. In June 2026 Federico Bianchi, Yongchan Kwon, Aneesh Pappu and James Zou described EinsteinArena, a platform on which AI agents submit solutions to public verifiers, and reported that agents raised the dimension-11 bound from 593 to 594 and, building on that result, to 604 [13]. A second version on 26 August was "Updated to correct a mistake", without saying which [13]. In early August 2026 OpenAI released "Ten Advances in Mathematics and Theoretical Computer Science", results it says were "obtained by an internal OpenAI model"; a remark in its chapter on codes states that its second-level certificates give τn ≤ 2^((0.39661+o(1))n), against 0.400944… for the optimised classical bound, and the chapter calls this the first improvement to the general exponent since 1978; the document was updated on 6 August [14]. On 28 September 2026 Shuxing Yang and colleagues reported new lower bounds in nineteen dimensions between 25 and 55, writing: "The autonomous system carried out the construction searches, mathematical analysis and computational verification" [15]. On 14 October 2026 Cohn's table credited its lower bounds in dimensions 11, 12, 18, 19, 21 and 25 to 46 to work posted in 2026, much of it as preprints or code repositories; the PackingStar and Ho bounds had been overtaken [9].
Main approaches
Linear and semidefinite programming (upper bounds). Find a certificate that limits the size of every configuration. Strongest evidence: the linear program is exact in dimensions 8 and 24 [3, 5], three-point bounds recover dimensions 3 and 4 [5], and de Laat and colleagues note that the Lasserre hierarchy "in principle" solves the problem in any dimension because it converges in finitely many steps [7]. Main difficulty: pure linear programming is provably too weak in dimension four [1], computing higher levels "is challenging" [7], and the semidefinite programs need high-precision arithmetic because of numerical instability [5, 6].
Constructions from lattices and codes (lower bounds). Take the shortest vectors of a lattice or the words of a code, then cut, lift or modify them. Strongest evidence: the E8 and Leech lattices give the optimal arrangements in dimensions 8 and 24 [5], and Ho's bound in dimension 19 comes from a binary code fed into Cohn and Li's "odd-sign" construction [12]. Main difficulty: above dimension 24 the records have been built by extending the Leech lattice arrangement [11], and the ratio between the bounds grows with dimension [9].
Numerical and AI-assisted search (lower bounds). Optimise point positions numerically, then make a nearly valid arrangement exact. Strongest evidence: the gains in dimension 11 from 592 to 604 [10, 13]; the EinsteinArena agents snapped near-integer inner products to exact values to obtain "a provably valid construction" [13]. Main difficulty: valid and invalid arrangements can differ by less than machine precision, so the EinsteinArena verifiers use 30 to 80 significant digits [13], and the Qiushi Engine authors write that turning such searches "into general mathematical constructions and rigorous proofs remains challenging" [15].
Random constructions in high dimensions. Build configurations from a random hard-sphere model. Strongest evidence: the factor-n and factor-3.44 improvements [8]. Main difficulty: these gains are polynomial and constant factors, and the exponential gap to 2^(0.401n) remains [1, 8].
Open sub-questions
- Dimension five. Is τ5 = 40, the lower bound [1, 9]? Can the second level of the Lasserre hierarchy, reported sharp in dimension four [7], bring the upper bound below 44?
- Dimension eleven. The arrangements from 594 to 600 share a backbone of 496 vectors [13]. Does that integral structure extend beyond 604, or can a semidefinite bound move below 868 [6, 9]?
- Rigorous three-point bounds. Do the bounds of Leijenhorst and de Laat in dimensions 11 to 23 survive the interval-arithmetic verification they did not perform [6]?
- Upper exponent. Does the claimed exponent 0.39661 hold, and do higher levels of the same certificates lower it further [14]?
- Lower exponent. Can any construction beat (2/√3)^n by an exponential factor, not only a polynomial one [1, 8]?
- Beyond the Leech lattice. In dimensions 25 to 31, can an arrangement not built around the Leech lattice beat the current records [9, 11]?
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Key references
- Boyvalenkov, P., Dodunekov, S. & Musin, O. (2012). A survey on the kissing numbers. Serdica Mathematical Journal 38, 507–522. Source · Checked 14 Oct 2026
- Schütte, K. & van der Waerden, B. L. (1953). Das Problem der dreizehn Kugeln. Mathematische Annalen 125, 325–334. 10.1007/BF01343127 · Checked 14 Oct 2026
- Odlyzko, A. M. & Sloane, N. J. A. (1979). New bounds on the number of unit spheres that can touch a unit sphere in n dimensions. Journal of Combinatorial Theory, Series A 26(2), 210–214. 10.1016/0097-3165(79)90074-8 · Checked 14 Oct 2026
- Musin, O. R. (2008). The kissing number in four dimensions. Annals of Mathematics 168(1), 1–32. 10.4007/annals.2008.168.1 · Checked 14 Oct 2026
- Bachoc, C. & Vallentin, F. (2008). New upper bounds for kissing numbers from semidefinite programming. Journal of the American Mathematical Society 21(3), 909–924 (published online 2007). 10.1090/S0894-0347-07-00589-9 · Checked 14 Oct 2026
- Leijenhorst, N. & de Laat, D. (2024). Solving clustered low-rank semidefinite programs arising from polynomial optimization. Mathematical Programming Computation 16(3), 503–534. 10.1007/s12532-024-00264-w · Checked 14 Oct 2026
- de Laat, D., Leijenhorst, N. M. & de Muinck Keizer, W. H. H. (2024). Optimality and uniqueness of the D4 root system. Preprint, arXiv:2404.18794 (version 2, 27 May 2024), not peer reviewed. Source · Checked 14 Oct 2026
- Gil Fernández, I., Kim, J., Liu, H. & Pikhurko, O. (2025). New lower bounds on kissing numbers and spherical codes in high dimensions. American Journal of Mathematics. 10.1353/ajm.2025.a966288 · Checked 14 Oct 2026
- Cohn, H. Kissing numbers. Table of the best known upper and lower bounds in dimensions 1 to 48, with references (no update date shown). Source · Checked 14 Oct 2026
- AlphaEvolve team, Google DeepMind (2025). AlphaEvolve: A Gemini-powered coding agent for designing advanced algorithms. Google DeepMind blog, 14 May 2025; white paper: Novikov, A. et al. (2025), AlphaEvolve: A coding agent for scientific and algorithmic discovery, arXiv:2506.13131. Source · Checked 14 Oct 2026
- Ma, C., Zhaowei, T. T., Li, P., Liu, M., Chen, H., Mao, Z. et al. (2025). Finding Kissing Numbers with Game-theoretic Reinforcement Learning. Preprint, arXiv:2511.13391 (version 1, 17 November 2025; version 5, 22 September 2026), not peer reviewed. Source · Checked 14 Oct 2026
- Ho, B. S. (2026). A new lower bound for the kissing number in 19 dimensions. Preprint, arXiv:2603.10425 (version 1, 11 March 2026; version 2, 20 March 2026), not peer reviewed. Source · Checked 14 Oct 2026
- Bianchi, F., Kwon, Y., Pappu, A. & Zou, J. (2026). Harnessing the Collective Intelligence of AI Agents in the Wild for New Discoveries. Preprint, arXiv:2606.10402 (version 1, 9 June 2026; version 2, 26 August 2026), not peer reviewed. Source · Checked 14 Oct 2026
- OpenAI (2026). Ten Advances in Mathematics and Theoretical Computer Science. Preprint PDF, updated 6 August 2026 (the original version is ten-proofs-oai-original.pdf in the same directory), not peer reviewed. Source · Checked 14 Oct 2026
- Yang, S., Zhao, R., Wu, J., Wang, Y., Chen, F., Zhu, K. et al. (2026). Large-Scale Autonomous Discovery of Kissing Number Constructions. Preprint, arXiv:2609.35051 (version 1, 28 September 2026), not peer reviewed. Source · Checked 14 Oct 2026
Source
Open, as listed by Wikipedia · Unsolved problems in mathematics. Checked 08 Oct 2026.
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Records on this problem
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