Result 373, Partial differential equations

Nonattainment of the three-marginal Coulomb Monge problem

An optimal three-particle Coulomb configuration need not be a deterministic function of the first particle, even for smooth identical spatial densities. The Monge and Kantorovich infima agree, but the Monge infimum is not attained. The same phenomenon occurs for every inverse-power Riesz exponent in each dimension at least two, with a suitable density in each case.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

For three repelling particles, the lowest interaction energy may be impossible to realize by making two particles' positions deterministic functions of the first. Smooth spatial densities do not, by themselves, guarantee this simpler description of an optimum.

What changes?

The manuscript reports a smooth, compactly supported probability density in three dimensions whose square root is also smooth and compactly supported. Giving each particle this density, no pair of Borel-measurable, distribution-preserving maps from the first particle attains minimum Coulomb energy. Yet maps can approach the minimum over all joint distributions. The same nonattainment and equality of infima hold for suitable smooth identical densities in every dimension at least two, for interactions summing pairwise reciprocal distances raised to any positive power.

What does that help mathematicians do?

The deterministic formulation is called the Monge problem; the Kantorovich formulation allows general joint probability distributions with the prescribed individual densities. This counterexample distinguishes matching optimal energy values from having equally expressive minimizers. A researcher cannot infer an optimal deterministic configuration merely because deterministic configurations approach the unrestricted minimum, even under the stated smoothness assumptions. It does not say deterministic minimizers always fail.

Are there practical applications?

The immediate value is foundational for transport problems with repulsive interactions. The result identifies a limitation of representing correlated particle positions through maps, while preserving their ability to approximate the optimal energy in these examples. It therefore separates energy approximation from exact optimizer representation, rather than providing a computational algorithm or a demonstrated practical application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A counterexample to the Monge ansatz for the three-marginal Coulomb cost

September 25, 2026 14 pages

We construct a smooth compactly supported probability density on ℝ3, with a smooth compactly supported square root, for which the three-marginal Coulomb transport minimum is not attained by any pair of measure-preserving Borel maps. Nevertheless, the Monge and Kantorovich infima agree: we construct preserving maps whose costs approach the Kantorovich minimum. For every d ≥ 2 and s > 0, we also construct a smooth identical marginal with the same nonattainment and equality-of-infima properties for the inverse-power interaction ∑i<j∣xi−xj∣−s\sum_{i\lt j}|x_i-x_j|^{-s} on ℝd.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026,
  author = {{OpenAI}},
  title = {{A counterexample to the Monge ansatz for the three-marginal Coulomb cost}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/paper.pdf}{OAI:A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/373.md.

Nonattainment of the three-marginal Coulomb Monge problem

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The Monge ansatz asks whether an optimal multi-marginal transport plan can be induced by maps from one marginal. For the three-marginal Coulomb cost in R3\mathbb R^3, the formalization constructs a smooth compactly supported probability density, with smooth compactly supported square root, for which no pair of measure-preserving Borel maps attains the Kantorovich minimum.

Nevertheless, the Monge and Kantorovich infima are equal: preserving maps with finite costs approach the minimum. This is the three-dimensional Coulomb result; the paper's inverse-power extensions in every dimension are outside this statement.

Comparator links

Result Comparator statement
Coulomb Monge nonattainment with equality of infima CoulombCounterexample.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.