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A counterexample to the Monge ansatz for the three-marginal Coulomb cost
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Abstract
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 $\sum_{i\lt j}|x_i-x_j|^{-s}$ on ℝd.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope note for family 373: Nonattainment of the three-marginal Coulomb Monge problem (lean/docs/373.md, 6 October 2026). The scope note says the formalization covers smooth-density nonattainment for the three-dimensional Coulomb problem and equality of the Monge and Kantorovich infima, and it links the comparator statement. It states that the paper's inverse-power Riesz extensions in other dimensions are outside the formalized statement. [family-level]
- OpenAI math release, CoulombCounterexample.lean (comparator challenge statement) (openai/math, 2026). This is the comparator statement
coulomb_counterexample_and_equal_infima, which asserts a density with nonattainment, equal infima and finite-cost Monge approximation. In this challenge file the statement is left assorry. - OpenAI math release, OAI.Analysis.CoulombTransport.Main (openai/math, 2026). This is the entry module of the solution development, and it imports
CoulombTransport.AllandCoulombTransport.Conclusion.
Possibly relevant literature (found by the agent, not checked)
- G. Friesecke, A simple counterexample to the Monge ansatz in multi-marginal optimal transport, convex geometry of the set of Kantorovich plans, and the Frenkel-Kontorova model (arXiv:1808.04318, 13 August 2018). This paper shows that the Monge ansatz fails for three marginals with symmetric pairwise (Frenkel–Kontorova) costs in a finite-state setting, and its continuous superposition exhibits nonattainment through microstructure. It is a possible comparison for the paper's Coulomb nonattainment and approximating-map construction.
- A. Moameni and B. Pass, Solutions to multi-marginal optimal transport problems concentrated on several graphs (arXiv:1507.05923, 2015). This paper studies multi-marginal optimizers concentrated on several graphs over the first marginal, and its examples include the one-dimensional Coulomb cost. It is related to the paper's optimizer, which is supported on four branch families and their permutations.
- B. Pass, Multi-marginal optimal transport: theory and applications (arXiv:1406.0026, 2014). This survey covers the existence, uniqueness and structure (including Monge form) of multi-marginal optimizers, the setting of the paper's Monge-attainment question.
Digests and expositions
- Gradient Damage, Boltzmann Equation, Hot Spots Conjecture and 7 More in Partial Differential Equations (video compilation, 15 October 2026). The chapter beginning at 2:06 introduces the three-particle Coulomb Monge problem and presents the release's nonattainment claim as an illustration rather than a proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 373
Independent sites; their content is not reviewed here.
Cite
@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 provides Lean files for this family: lean/docs/373.md. The Lean statements have not been checked by people against the paper's claims.
Family 373: 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.
This is the only paper in this family.