Simon's Problem 7 (1984): a continuum phase transition for a stable tempered pair potential, on an open interval of densities

Fix a pair potential $v$ on $\mathbb R^3$ that is stable, $\sum_{i<j}v(x_i-x_j)\ge-CN$ , and tempered, $|v(x)|\le C(1+|x|)^{-3-\epsilon}$ . For $N$ particles in a volume $\Lambda$ at inverse temperature $\beta$ , let $Z_\Lambda=\int\prod d^3p_i\,d^3x_i\,e^{-\beta H}$ with $H=\sum p_i^2/2+\sum_{i<j}v(x_i-x_j)$ ; as $\Lambda\to\mathbb R^3$ with $N/|\Lambda|\to\rho$ , $|\Lambda|^{-1}(-\ln Z_\Lambda)$ converges to a function $f(\beta,\rho)$ concave in $\beta$ . A first-order phase transition is a failure of $f$ to be $C^1$ in $\beta$ . Rigorous transitions were known for lattice systems and only one rather artificial continuum model. Simon's Problem 7: show that for suitable choices of $v$ , and for $\rho$ sufficiently large, $f$ is non- $C^1$ at some $\beta$ .

Result: Proved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Construction
Field: Classical statistical mechanics; continuum phase transitions
Posed by: Barry Simon (Problem 7, Fifteen problems in mathematical physics)
Year posed: 1984
Years open: 42y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 38 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Algebraic-decay manuscript, Theorem 1.1: there is a bounded continuous radial $\phi$ on $\mathbb R^3$ with $\sum_{i<j}\phi\ge-BN$ and $|\phi(r)|\le Cr^{-3-1/32}$ ( $r\ge1$ ), an open interval $I$ centred at $5p/3$ ( $p$ the unit packing density) and one $\beta_c\in[7/8,9/8]$ such that for every $\rho\in I$ the canonical free energy exists for all $\beta>0$ and $f'_-(\beta_c,\rho)>f'_+(\beta_c,\rho)$ . The companion gives a potential with divergent core and $o(r^{-3})$ tail, a common $\beta_c\in(1/2,3/2)$ on an open density interval, and says it does not meet a fixed power margin. Not shown: a transition for every sufficiently large $\rho$ as Simon's wording asks, for Lennard-Jones or any finite-range potential, or identification of the phases. Earlier continuum transitions (Widom-Rowlinson, Lebowitz-Mazel-Presutti, recent Kac-type models) used several species, many-body terms or box-dependent interactions.

What the AI did

The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts dated September 24, 2026: one with a divergent repulsive core and an $o(r^{-3})$ tail, and one with a bounded continuous potential and an explicit power tail. Both have Lean formalizations of their main theorems.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the algebraic-decay manuscript was read against Simon's Problem 7 as printed in 1984. The potential is bounded, continuous, stable and satisfies $|\phi(r)|\le Cr^{-3-1/32}$ for $r\ge1$ , so it meets Simon's stability and temperedness conditions; Simon's kinetic term only adds a smooth function of $\beta$ . Lean: lean/ComparatorChallenges/RadialDensityInterval.json exists with solution module OAI.Probability.RadialTransition.DensityInterval present at the pinned commit; it is not listed in lean/formalization.yaml. Its statement OAI.RadialTransition.densityInterval was read here and states the headline (bounded continuous stable potential with that decay, a density interval around $5p/3$ , one common $\beta_c\in[7/8,9/8]$ with a strict corner of the canonical free energy). Not rebuilt here. The companion's ContinuumTransition and RadialTransition statements are in the catalogue. The papers say the potential is engineered and is not Lennard-Jones, and do not identify the phases.

Simon's Problem 7 (1984): a continuum phase transition for a stable tempered pair potential, on an open interval of densities

What was actually shown

What the AI did

Verification

Sources

Discussion