Result 228, Probability and statistical mechanics

Continuum phase transitions for radial pair potentials

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 formalization Proof

The bigger picture

Why it matters

Particles moving freely in space can, according to these manuscripts, undergo an abrupt thermodynamic transition even when their interactions depend only on distance. The constructions provide concrete mathematical examples of discontinuous temperature response.

What changes?

The manuscripts construct stable, distance-dependent pair potentials in three dimensions. One has a divergent repulsive core, an attractive interval, and an integrable tail decaying faster than inverse distance cubed. The other is bounded and continuous, with absolute value at most C times distance to the power minus 3 minus 1/32 for distances at least one. For each, the canonical free-energy derivative jumps strictly downward at one common finite positive inverse temperature throughout an open interval of positive densities.

What does that help mathematicians do?

Stability means total energy is bounded below by a fixed negative constant times particle number. The canonical free energy describes equilibrium at fixed density; both constructions keep it finite at every positive density and inverse temperature. Thus the claimed transition is a jump in its slope with respect to inverse temperature, not a divergence of free energy. The bounded example shows that an infinitely strong short-distance repulsion is not required for this phenomenon.

Are there practical applications?

The immediate value is foundational for statistical mechanics: these are continuum models with a specified temperature singularity, rather than models restricted to lattice sites. They realize the type of transition sought in Simon's continuum problem. The claims concern specially constructed interactions, not predictions for a particular material or every radial potential.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

A continuum temperature singularity for a radial pair potential

September 24, 2026 31 pages Main result formalized in Lean

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.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026,
  author = {{OpenAI}},
  title = {{A continuum temperature singularity for a radial pair potential}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf}{OAI:A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026}},
  year = {2026}
}

A radial continuum phase transition with algebraic decay

September 24, 2026 33 pages Main result formalized in Lean

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.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026,
  author = {{OpenAI}},
  title = {{A radial continuum phase transition with algebraic decay}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf}{OAI:A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/228.md.

Continuum phase transitions for radial pair potentials

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization constructs one stable radial pair potential in three-dimensional continuum space with a divergent repulsive core and an integrable tail that is $o(r^{-3})$ . Its canonical free energy exists for every positive inverse temperature and density. At one inverse temperature $\beta_c\in(1/2,3/2)$ , the free energy has a strict finite jump between its one-sided derivatives with respect to inverse temperature, for every density in one nonempty open interval.

The formalization constructs a bounded continuous stable radial pair potential in $\mathbb R^3$ satisfying $|\phi(r)|\le Cr^{-3-1/32}$ for $r\ge1$ . There is a nonempty open interval of positive densities around $5p/3$ , where $p$ is the unit-separated packing-density limit, on which the canonical free energy exists at every positive inverse temperature and has a strict downward derivative jump at one common $\beta_c\in[7/8,9/8]$ .

The earlier fixed-density statement at $5p/3$ is also retained. Both statements concern the derivative with respect to inverse temperature.

Comparator links

Result	Comparator statement
Temperature singularity over a density interval	ContinuumTransition.lean
Fixed-density radial continuum phase transition	RadialTransition.lean
A common radial phase transition over a density interval	RadialDensityInterval.lean

At a glance

Subject: Probability and statistical mechanics
Manuscripts: 2
Pages: 64
Dated: Sep 24, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Simon’s continuum problemProved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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