Can one kind of particle provably change phase?
Two preprints claim a temperature singularity for a gas with a radial pair potential
Covers result 228 in OpenAI’s catalog: all 2 papers. See its connections to other results and the work it cites.
4:26. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Put identical particles in space and let every pair interact through the same function of their distance. Can such a gas show a sharp phase transition? On lattices, rigorous transitions go back to Peierls in the 1930s; in the continuum the question resisted. In 1984 Barry Simon listed it as Problem 7 of his fifteen problems in mathematical physics: find a stable pair potential, bounded and decaying faster than r−3 by a power, for which, at large densities, the free energy is not C1 in temperature. Most earlier proofs needed props: two species (the Widom–Rowlinson model), four-body forces (Lebowitz, Mazel and Presutti), or a pair interaction tied to a fixed grid of boxes (2025). In 1975 Israel got two coexisting densities, but with a hard core that forbids a whole range of distances.
Result 228 is two OpenAI preprints, 64 pages, dated September 24, 2026. Each constructs a stable radial pair potential in three dimensions, an open interval of densities, and one inverse temperature at which the temperature derivative of the canonical free energy jumps, for every density in the interval. In the first paper the repulsion diverges at zero distance; in the second the potential is bounded and continuous, with a tail bounded by C·r−3−1/32. The second seems to meet Simon’s literal conditions, though for one interval of densities rather than all large ones; both papers cite Simon’s list, but neither names Problem 7 or claims to solve it.
What it doesn’t claim
The Lennard-Jones potential, phases identified as liquid or vapor, values for the constants, or a natural-looking potential. The potentials are engineered: by our reckoning from one paper’s stated conditions, with its example kernel, the core parameter must exceed about 300,000, putting the special shell below 10−131,428.
Where it stands
- Posted as preprints
- OpenAI, two papers, 64 pages, dated September 24, 2026
- Peer reviewed
- Not yet
- Checked in Lean
- Yes (docs)
Last checked October 19, 2026. Neither preprint has been revised since release. In Lean, three comparators cover both theorems, about 265 files with no unproved steps (we did not build them); one comparator is missing from the formalization list. Neither paper is among those OSVerify reviewed, and we found no expert reaction and no competing claim.
How the proof goes
- A core that allows only tetrahedra. Distances below one cost a large energy, except in a hair-thin shell at almost zero distance. No five points can all sit at shell distances from each other, so at most four particles can huddle. A counting bound from graph theory (Turán) then gives stability. Three ways to fill space compete: empty space, separated single particles, and tetrahedral clusters.
- A triple point. One ideal mean-field attraction of the right strength makes the three options meet at a single point, at inverse temperature 1. At fixed density, passing through that temperature switches which two phases coexist (singles and clusters when hotter, empty space and clusters when colder), releasing latent heat: the jump in the derivative.
- Infinitely many attractions. A real potential could round that corner off, so the papers add infinitely many ever-weaker, ever-longer-range attractions, each chosen to protect the corner from the errors of the last.
The numbers in the video
- Ours: the papers’ ideal model computed exactly: the triple point (β, h) = (1, −3/4), and at density 5p/3, where p is the packing density, a jump of 1.75p in the scaled energy. Five points need a distance ratio of at least √(12/7) ≈ 1.309 (a triangular bipyramid), far above the shell’s 1.001.
- Ours: a cartoon Monte Carlo with 32 particles and a schematic bounded potential; on cooling, the particles gather into four-particle clusters. At that size the change is smooth, as it must be. The core-parameter bound is our arithmetic from one paper’s conditions.
Read the source
- A continuum temperature singularity for a radial pair potential (PDF), 31 pages.
- A radial continuum phase transition with algebraic decay (PDF), 33 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 228.
The 2 papers in this result
OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.
Transcript
Read the full script
0:00The question
Put identical particles in space, and let every pair interact through the same function of their distance. Can such a gas show a sharp phase transition, such as water boiling? On lattices, rigorous transitions go back to Peierls in the 1930s. In the continuum, the question resisted. In 1984, Barry Simon listed it as Problem 7 of fifteen in mathematical physics: find a stable pair potential, bounded, and decaying faster than distance^(−3) by a power, for which, at large densities, the free energy has a kink in temperature. Most proofs needed props: two species of particles, as in the Widom–Rowlinson model; four-body forces, in work of Lebowitz, Mazel and Presutti; or a pair interaction tied to a fixed grid of boxes, in 2025. In 1975, Israel got two coexisting densities, but only with a hard core that forbids a whole range of distances. And for realistic potentials, like Lennard-Jones, experts wrote in 2025 that the situation had remained largely unchanged for a quarter century.
1:11The claim
Result 228 is two OpenAI preprints, 64 pages, dated September 24, 2026. Each constructs a stable radial pair potential in three dimensions, with an open interval of densities and one inverse temperature at which the temperature derivative of the canonical free energy jumps, for every density in the interval. In the first paper the repulsion diverges at zero distance. In the second the potential is bounded and continuous, with a tail bounded by distance^(−3−1/32). The second seems to meet Simon’s literal conditions, though for one interval of densities rather than all large ones, and though both cite Simon’s list, neither names Problem 7 or claims to solve it. Not claimed: the Lennard-Jones potential, phases identified as liquid or vapor, values for the constants, or a natural-looking potential. These potentials are engineered: by our reckoning from one paper’s stated conditions, with its example kernel, the core parameter must exceed about 300,000, putting the special shell below 10^(−131,000).
2:23How the proof goes
The trick starts with the core. Distances below one cost a large energy, except in a hair-thin shell at almost zero distance. No five points can all sit at shell distances from each other, so at most four particles can huddle: a tiny tetrahedron. A counting bound from graph theory then gives stability. With this core, three ways to fill space compete: empty space, separated single particles, and tetrahedral clusters. Add one ideal, mean-field attraction of the right strength, and the three options meet at a single point, a triple point, at inverse temperature one. At fixed density, passing through that temperature switches which two phases coexist, releasing latent heat: the jump in the derivative. A real potential could round that corner off, so the papers add infinitely many ever-weaker, ever-longer-range attractions, each chosen to protect the corner from the errors of the last. As an illustration, our computer worked out the papers’ ideal model exactly: the triple point at (1, −3/4), and at density 5p/3, where p is the packing density, a jump of 1.75p in the scaled energy. A cartoon simulation with 32 particles shows them gathering into four-particle clusters as the temperature drops, though at that size the change is smooth, as it must be.
3:44Where it stands
Both preprints are unrefereed and unchanged since release. In Lean, three comparators cover both theorems, about 265 files with no unproved steps; we did not build them, and one comparator is missing from the formalization list. Neither paper is among those OSVerify reviewed, and we found no expert reaction and no competing claim. If they hold, one kind of particle with a radial pair potential, finite at every distance, can show a temperature singularity at fixed density in three dimensions, though the potential is engineered, far from any real gas.
Spot a mistake?
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