A computer-assisted proof of Kuperberg's six-cylinder conjecture

Ivan Matić; Rados Radoičić

A computer-assisted proof of Kuperberg's six-cylinder conjecture

Combinatorics 2026-07-27 v1

Authors: Ivan Matić , Rados Radoičić

Abstract

We prove that at most six pairwise non-overlapping infinite unit cylinders can simultaneously touch a unit ball. Six is achievable, so the answer to Kuperberg's question is exactly six. The proof is computer-assisted in the following sense: the statement is reduced to a finite case analysis with 2,954,984 cases, and in each case a program verifies that one of three inequalities between explicit polynomials with rational coefficients holds throughout a box. Each case is an elementary arithmetic check, and the complete list of cases is provided with the paper.

Keywords

Kuperberg's conjecture unit cylinders unit ball computer-assisted proof finite case analysis

Cite

@article{arxiv.2607.24691,
  title  = {A computer-assisted proof of Kuperberg's six-cylinder conjecture},
  author = {Ivan Matić and Rados Radoičić},
  journal= {arXiv preprint arXiv:2607.24691},
  year   = {2026}
}