Preprint: Reinforcement learning yields improvements to 15 longstanding kissing-number and spherical-code bounds

A research team led by Peking University says it has used reinforcement learning to make a rare kind of advance in pure mathematics: improving 15 longstanding lower bounds for kissing numbers and related spherical-code problems, a class of geometry questions that has challenged mathematicians for centuries. But the work is still an arXiv preprint, not a peer-reviewed journal paper, and the reported results do not all rest on the same level of proof.

The paper, “Finding Kissing Numbers with Game-theoretic Reinforcement Learning,” was first posted to arXiv on Nov. 17, 2025, and its current version, v5, appeared Sept. 22, 2026. Lead author Chengdong Ma lists affiliations with Peking University’s Institute for Artificial Intelligence, the Shanghai Academy of AI for Science and Fudan University’s AI institute. The authors describe their system, called PackingStar, as a reinforcement-learning method that recasts the search as a two-player cooperative matrix-completion game, working with Gram, or cosine, matrices rather than directly placing points in space.

The headline mathematical claims are in dimensions where exact answers are still unknown. For the classical kissing-number setting, corresponding to an inner-product threshold of α = 1/2, the paper reports improved lower bounds in dimensions 25 through 31. In dimension 25, for example, it gives 197,056, up from 197,048. In dimension 31, it reports 238,350, up from 232,874. The paper also says it found a new rational construction in 13 dimensions at α = 1/2, raising the lower bound to 1,146 from 1,130.

Other improvements listed in the paper include 252 in dimension 14 at α = 1/3, up from 240; 578 in dimension 17 at α = 1/3, up from 546; 704 in dimension 18 at α = 1/3, up from 672; 81 in dimension 12 at α = 1/4, up from 79; 405 in dimension 20 at α = 1/4, up from 378; 567 in dimension 21 at α = 1/4, up from 554; and 352 in dimension 22 at α = 1/5, up from 338.

The authors say the work goes beyond a numerical search. The paper reports exact three-point semidefinite-programming certificates for some cases, and its appendix gives exact integer upper bounds matching discovered configurations for several of them, including 12:81, 20:405, 21:567 and 22:352. In those instances, the evidence is stronger because the claimed construction is paired with an exact upper bound showing no larger code exists under the stated conditions.

That level of rigor does not yet extend uniformly across the whole paper. Many of the higher-dimensional configurations released by the authors are floating-point coordinate files checked numerically with tolerances rather than by exact arithmetic. That is common in computational search papers, but it is weaker than a full exact proof. One point already emerging in discussion of the preprint is the distinction between results backed by exact certificates and those now supported mainly by numerical verification.

The kissing number problem dates at least to Isaac Newton in 1694 and asks how many non-overlapping unit spheres can touch a central unit sphere in a given dimension. Exact kissing numbers are known only in a small set of dimensions commonly cited in the literature: 1, 2, 3, 4, 8 and 24. That is why even modest lower-bound improvements matter in discrete geometry, coding theory and information theory.

In the abstract, the authors write that “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.” The paper also states an upper bound of 142,155 for a specific code in S^77 and says, “Equality is attained by the Fi22 code,” which the authors present as the first explicit spherical-code realization of the Fischer sporadic group Fi22.

The team has released data and verification code on GitHub, including coordinate files, Gram matrices and scripts to check them. In the repository README, the authors say, “All of these configurations have been independently verified and have been accepted into the kissing number database and spherical codes database.” As of Sept. 23, 2026, however, the work remains a public preprint awaiting broader independent mathematical vetting and peer review.