The math of prime numbers and the physics of 3D shapes are actually the exact same language.
April 14, 2026
Original Paper
A Neukirch-Uchida Theorem for 3-Manifolds
arXiv · 2604.09469
The Takeaway
This proof shows that you can determine the exact shape of a complex 3D space just by looking at a specific map of its infinite links. It unites topology and number theory—two of the most distant branches of mathematics—into a single framework.
From the abstract
The classical Neukirch-Uchida theorem states that the absolute Galois group determines a number field up to isomorphism. We prove an analogue of this theorem for 3-manifolds in the framework of arithmetic topology. We study infinite links in 3-manifolds that behave like the set of primes, satisfying a Chebotarev density property. Relative to such a stably Chebotarev link, we define the absolute Galois group of a 3-manifold as the inverse limit of profinite completions of finite sublink complemen