Large-Scale Autonomous Discovery of Kissing Number Constructions

2026-09-28

AI Summary

This study addresses the challenge of translating AI-driven search into generalizable mathematical constructions and rigorous proofs for high-dimensional kissing number problems. We propose an autonomous multi-agent system built upon the Qiushi Engine, integrating techniques such as coordinated motion layers, symbolic code substitution, lattice isomorphisms, and spherical design moment certificates. This framework advances beyond fixed-formula optimization to discover novel mathematical constructions, generating independently verifiable finite certificates that overcome the limitations of traditional numerical searches. Our approach establishes new lower bounds on kissing numbers across 19 dimensions—for instance, K(55) ≥ 53,301,140—while determining sharp capacities and exact projection counts. All findings are substantiated through rigorous mathematical argumentation.

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, $K(25)\ge197580$, $K(27)\ge201567$, $K(38)\ge591900$, $K(43)\ge2553792$, $K(45)\ge7380090$, and $K(55)\ge53301140$. 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.

Problem

• kissing number
• discrete geometry
• mathematical construction
• lower bounds
• autonomous discovery

Innovation

• autonomous multi-agent system
• kissing number
• mathematical construction
• verifiable certificates
• discrete geometry