Large-Scale Autonomous Discovery of Kissing Number Constructions
Abstract
The kissing-number problem is a classical problem in discrete geometry whose exact solution is known in only a few dimensions. Recent artificial-intelligence approaches have begun to discover improved configurations through large-scale numerical and combinatorial search, but converting such searches into general mathematical constructions and rigorous proofs remains challenging. 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 $49−−55$. The resulting constructions arise from distinct structural mechanisms, including coordinated motions of contact layers, labelled direction reuse, joint support exchanges, signed-code replacements, cross-shell lattice constructions, low-overlap lattice isometries, and spherical-design moment certificates. They yield sharp capacities for parameterized signed-code models, deterministic image-union guarantees, and exact section and projection counts controlled by embedded root systems and anchor-graph statistics. These methods yield, among others, K(25)≥197580, K(27)≥201567, K(38)≥591900, K(43)≥2553792, K(45)≥7380090, and K(55)≥53301140. 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. Our results illustrate how autonomous research systems can move beyond optimization within fixed formulations to discover new mathematical representations and constructions at scale.
Related
- Lattices with Large Kissing Number from Local Density
- On the kissing number of hyperbolic surfaces with a small systole
- Improved lower bounds for the kissing number in dimensions 49 to 61
- NeuralCert: certified computational discovery of extremal mathematical constructions
- Towards Learning-Guided Discovery of Extremal Geometric Constructions: A PatternBoost Study
BibTeX
@misc{yang2026largescale,
title = {{Large-Scale Autonomous Discovery of Kissing Number Constructions}},
author = {Shuxing Yang and Rui Zhao and Junyao Wu and Yize Wang and Fujia Chen and Kaihao Zhu and Wenhao Li and Zichen Li and Yaqi Li and Shenzhan Hong and Yuang Pan and Junjie Yang and Taowen Deng and Jincheng Mi and Hongsheng Chen and Yihao Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35051},
eprint = {2609.35051},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35051},
}
RIS
TY - UNPB
TI - Large-Scale Autonomous Discovery of Kissing Number Constructions
AU - Shuxing Yang
AU - Rui Zhao
AU - Junyao Wu
AU - Yize Wang
AU - Fujia Chen
AU - Kaihao Zhu
AU - Wenhao Li
AU - Zichen Li
AU - Yaqi Li
AU - Shenzhan Hong
AU - Yuang Pan
AU - Junjie Yang
AU - Taowen Deng
AU - Jincheng Mi
AU - Hongsheng Chen
AU - Yihao Yang
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35051
UR - https://arxiv.org/abs/2609.35051
AB - The kissing-number problem is a classical problem in discrete geometry whose exact solution is known in only a few dimensions. Recent artificial-intelligence approaches have begun to discover improved configurations through large-scale numerical and combinatorial search, but converting such searches into general mathematical constructions and rigorous proofs remains challenging. 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 $49−−55$. The resulting constructions arise from distinct structural mechanisms, including coordinated motions of contact layers, labelled direction reuse, joint support exchanges, signed-code replacements, cross-shell lattice constructions, low-overlap lattice isometries, and spherical-design moment certificates. They yield sharp capacities for parameterized signed-code models, deterministic image-union guarantees, and exact section and projection counts controlled by embedded root systems and anchor-graph statistics. These methods yield, among others, K(25)≥197580, K(27)≥201567, K(38)≥591900, K(43)≥2553792, K(45)≥7380090, and K(55)≥53301140. 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. Our results illustrate how autonomous research systems can move beyond optimization within fixed formulations to discover new mathematical representations and constructions at scale.
ER -