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A continuum temperature singularity for a radial pair potential
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Abstract
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.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Continuum phase transitions for radial pair potentials (Lean scope notes) (lean/docs/228.md, 2026). Release's scope note: it describes the formalized statement of this paper's theorem (divergent core, $o(r^{-3})$ tail, a derivative jump at one $\beta_c\in(1/2,3/2)$ on an open density interval) and links the comparator file
ContinuumTransition.lean. [family-level] - OpenAI, ContinuumTransition/Main.lean (openai/math, commit adc7f12). The release's Lean module that proves
main_theorem : MainStatement, the formal counterpart of this paper's Theorem 1.1.
Possibly relevant literature (found by the agent, not checked)
- Göbel, Jenssen, Michelen, Pappik, Perkins, Schiller, Uniqueness, analyticity and mixing for Gibbs point processes via spectral gaps (arXiv:2606.28009, June 2026). Shows that for certain repulsive radial pair potentials there is no phase transition at any activity; this contrasts with the paper's construction, which adds attractive radial terms to obtain a temperature singularity.
Elsewhere
- Emergent Mind: family 228
Independent sites; their content is not reviewed here.
Cite
@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}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/228.md. The Lean statements have not been checked by people against the paper's claims.
Family 228: 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.
Other papers in this family:
- A radial continuum phase transition with algebraic decay · September 24, 2026