Continuum phase transitions for radial pair potentials

What OpenAI claims

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)

Papers (2)

1. A continuum temperature singularity for a radial pair potential

   24 September 2026 · Preprint, not peer reviewed · PDF

   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 ϕ(r)=o(r^-3). Its free-cube canonical free energy is finite at every positive inverse temperature and density.

2. A radial continuum phase transition with algebraic decay

   24 September 2026 · Preprint, not peer reviewed · PDF

   We construct a bounded, continuous, stable radial pair potential in three dimensions with |ϕ(r)| ≤ 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.