1 · The essentials
These manuscripts construct two mathematical models of particles moving in three-dimensional space. In each model, the interaction between two particles depends only on their distance apart. Both models exhibit a sharp change: a quantity describing the system’s equilibrium behavior remains finite, but its slope suddenly drops as inverse temperature passes a particular value. Within each model, that same value works across a whole open range of positive particle densities, rather than just one density. One construction makes extremely close particles repel increasingly strongly, while the other keeps the interaction bounded and continuous at all distances. Both main transition results have Lean formalizations.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, picture a road whose elevation stays finite but whose slope abruptly changes at a bend. The transition here concerns a similar change in slope, except that the horizontal coordinate is inverse temperature and the measured quantity describes a many-particle system.
Why it matters
The constructions give continuum particle models with distance-only interactions that exhibit a first-order phase transition in the precise sense of a derivative jump. They achieve this with two different kinds of short-distance behavior, including a bounded, continuous interaction. The repository summary identifies the bounded construction as realizing the type of transition sought in Simon’s continuum problem.
2 · Going further
The problem is to construct a stable radial pair potential for classical particles in three dimensions whose canonical free energy has a temperature singularity. Writing inverse temperature as and density as , the asserted feature is a strict downward jump in the derivative with respect to at a finite , for every in an open interval of positive densities. Within each construction, is common to that entire interval; the source does not say that the two constructions share the same value.
The main manuscript constructs a potential with a divergent repulsive core, a nontrivial attractive interval, and an integrable tail satisfying . Its free-cube canonical free energy is finite for every positive inverse temperature and every positive density, while the derivative-jump conclusion applies throughout the specified open density interval.
The companion manuscript constructs a bounded, continuous, stable radial potential satisfying for . Throughout an open interval of positive densities, its canonical thermodynamic free energy is finite at every positive inverse temperature and has the same kind of strict downward derivative jump at one common finite positive inverse temperature. Thus it supplies an alternative construction with bounded short-distance behavior and an explicit algebraic decay bound.
The abstracts describe the approach as constructing suitable potentials, but do not explain the mechanism or intermediate steps of the argument. Both manuscripts are marked Lean-formalized. The supplied text does not specify the exact coverage of those formalizations beyond identifying the main results as formalized.
Useful notions
- Particle density
- Temperature and inverse temperature
- Functions and their derivatives
- Interactions depending on distance
- Equilibrium free energy
What was known before / what is new
Before
The repository summary connects this family to Simon’s continuum problem and says the bounded construction realizes the type of transition sought there. The supplied text gives no further account of earlier results or of the full formulation of that problem.
New
The main manuscript constructs a stable radial interaction with a divergent repulsive core and a free-energy derivative jump at a density-independent inverse temperature over an open density interval. The companion achieves the same type of singularity using a bounded, continuous potential with an explicit integrable power-law decay bound. Both main results have Lean formalizations.
Limits and assumptions
These are constructions of particular stable radial pair potentials in three dimensions, not a claim about every distance-dependent interaction. The transition is asserted on an open interval of positive densities, not at all positive densities. The main manuscript states free-energy finiteness at all positive inverse temperatures and densities; the companion states it throughout its specified density interval at all positive inverse temperatures. Both main results are Lean-formalized, but the supplied abstracts do not delineate the formalization of every subsidiary claim. They also do not identify remaining open questions or provide enough detail to assess the full scope of Simon’s continuum problem.
Mini glossary
- Radial pair potential —
- An interaction between two particles that depends only on the distance separating them.
- Inverse temperature —
- The temperature parameter whose variation is used to locate the derivative jump in these results.
- Canonical free energy —
- The equilibrium quantity studied at specified inverse temperature and particle density.
- Strict downward derivative jump —
- An abrupt decrease in a function’s slope when its input passes a specified value.
- Divergent repulsive core —
- Short-distance behavior in which the repulsion grows without bound as particles approach one another.
- Algebraic decay —
- A decrease with distance controlled by a negative power of that distance, as in the companion’s stated tail bound.
Keywords : Radial pair potentials · Continuum phase transitions · Canonical free energy
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A continuum temperature singularity for a radial pair potential
24/09/2026We 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 . Its free-cube canonical free energy is finite at every positive inverse temperature and density.
A radial continuum phase transition with algebraic decay
24/09/2026We construct a bounded, continuous, stable radial pair potential in three dimensions with 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.