If you look at every whole number in existence, they actually act exactly like a cloud of gas following the laws of physics.
April 2, 2026
Original Paper
Statistical Physics of Coding for the Integers
arXiv · 2604.00858
The Takeaway
Researchers found that when you try to compress whole numbers into computer code, they mirror the physics of a 'Bose gas,' where energy levels are dictated by prime numbers. The math suggests that numbers aren't just abstract concepts, but follow a physical 'boiling point' (a Hagedorn transition) where the laws of information storage fundamentally break down.
From the abstract
We study a paradigm of coding for compression of the natural numbers via the zeta distribution and develop a statistical-mechanical interpretation, both in terms of Hagedorn systems and a Bose gas with energy levels given by logarithms of prime numbers. We also propose a simple coding scheme for the zeta distribution that nearly achieves the ideal code length. For block coding of vectors of natural numbers, we derive the micro-canonical entropy function and demonstrate its asymptotic linearity i