Constructs stable distance-dependent pair interactions for three-dimensional classical particles with a first-order phase transition: the canonical free energy has a derivative jump at one inverse temperature throughout an open density interval. One potential has a divergent repulsive core; another is bounded and continuous with an integrable power-law tail, realizing the type of transition sought in Simon's continuum problem. (Lean)
Manuscripts (2)
A continuum temperature singularity for a radial pair potential
September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026}
We construct a stable radial pair potential in three dimensions whose canonical free energy has a strict downward derivative jump at one common finite positive inverse temperature throughout an open interval of positive densities. The potential has a divergent repulsive core, a nontrivial attractive interval, and an integrable tail satisfying \phi(r)=o(r^{-3}). Its free-cube canonical free energy is finite at every positive inverse temperature and density.
A radial continuum phase transition with algebraic decay
September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026}
We construct a bounded, continuous, stable radial pair potential in three dimensions with |\phi(r)|\le Cr^{-3-1/32} for r ≥ 1. Throughout an open interval of positive densities, its canonical thermodynamic free energy is finite at every positive inverse temperature and has a strict downward derivative jump at one common finite positive inverse temperature.
Result 228 · Probability and statistical mechanics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md