Characterization of the limiting solution in Smale’s seventh problem

Characterize the solution of the interacting-particle energy problem in the limit s\to 0, as posed by Smale’s seventh open problem.

Background

The paper studies an interacting energy defined over transformer word embeddings, consisting of a quadratic attractive term and a repulsive inverse-power term parameterized by s. The authors relate this energy to particle systems whose minimizers become uniformly distributed on a sphere in the logarithmic limit s\to 0. In this context, they identify the characterization of the corresponding limiting solution as Smale’s seventh open problem. The question is included as mathematical background for the energy-based interpretation of in-context sampling, rather than as a problem resolved by the transformer constructions in the paper.

References

For example, Smale's 7th open problem aims at characterizing the solution for $s\to 0$.

— Transformers as In-Context Samplers: From Closed-Form Diffusion to Estimation-Free Sampling  (2609.08981 - Adibi et al., 8 Sep 2026) in Section 5.1, “Observation: Embeddings as Interacting Particles”