View source for The Monge ansatz for the three-marginal Coulomb cost

'''The Monge ansatz for the three-marginal Coulomb cost''' asks whether, for three particles with the same absolutely continuous spatial density and the Coulomb repulsion between them, the optimal arrangement can always be taken deterministic: the position of one particle determines the positions of the other two through measure-preserving maps. This co-motion ansatz is used in the strong-interaction limit of density functional theory. A manuscript dated 25 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]) claims a smooth density in three dimensions for which no optimal map exists, although maps come arbitrarily close to the optimal value; the collection includes a Lean formalization of the three-dimensional Coulomb statement. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the manuscript:
** The co-motion formulation was developed in the strictly correlated electron literature of Seidl, Gori-Giorgi and Savin, and is surveyed by Friesecke, Gerolin and Gori-Giorgi. Colombo and Di Marino proved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic maps for identical non-atomic marginals, which gives approximating maps but not a minimizing map.
** For two particles, Cotar, Friesecke and Klüppelberg proved existence and uniqueness of an optimal map for absolutely continuous marginals in every dimension. On the line, Colombo, De Pascale and Di Marino constructed optimal cyclic maps for any number of particles. Pass constructed Coulomb optimizers not induced by maps for radial data, which does not exclude another optimizer of Monge form; Colombo and Stra, and Bindini, De Pascale and Kausamo, showed that particular prescribed radial maps are not optimal.
** Friesecke stated the three-particle Coulomb question in three dimensions as open in January 2026. Gerolin, Kausamo and Rajala had found a splitting mechanism for the repulsive harmonic cost, which the manuscript adapts.

== The original result ==
* '''Claim.''' Theorem 1.1 of ''A counterexample to the Monge ansatz for the three-marginal Coulomb cost'' states that there is a nonnegative density ρ on '''R'''<sup>3</sup>, smooth and compactly supported together with its square root, such that the three-marginal Coulomb transport minimum is finite and attained by a coupling, but every pair of Borel maps pushing the density forward to itself gives a strictly larger cost. The manuscript also states that the Monge and Kantorovich infima are nevertheless equal, and it extends both statements to every inverse-power interaction with exponent ''s'' &gt; 0 in every dimension ''d'' ≥ 2, with a suitable density in each case. With the particle normalization, the density lies in the physical density class used by Friesecke, Gerolin and Gori-Giorgi; no external potential or ground state is asserted.
* ''Method, as described:'' the marginal has five components, a central one with a third of the mass near the origin and four with a sixth each near ±''e''<sub>1</sub> and ±''e''<sub>2</sub>. The triples joining the origin to ±''e''<sub>1</sub> and to ±''e''<sub>2</sub> have the same cost and the same central gradient, so local calculus gives two families of triples with a common supporting potential, and every optimal plan is concentrated on them. A deterministic map would have to choose one branch on each set of central points and would send a third of that mass into a component image that carries only a sixth, which contradicts preservation of the marginal. Finer and finer cell decompositions give measure-preserving maps whose costs approach the minimum.
* '''Source''' (openai/math repository, commit <code>7ef5c0c</code>): [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026 A counterexample to the Monge ansatz for the three-marginal Coulomb cost], dated 25 September 2026.
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/lean/docs/373.md lean/docs/373.md] states that the formalization constructs the three-dimensional Coulomb counterexample and proves the equality of the Monge and Kantorovich infima, and that the inverse-power extensions in every dimension are outside this statement. The Comparator statement is <code>CoulombCounterexample.lean</code>.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 29 October 2026)
|-
| 2026-10-06 || [https://lowrank.github.io/latent-seminar/OpenAI-Math/373-coulomb-monge-nonattainment ''Nonattainment of the three-marginal Coulomb Monge problem''], an article of the Latent Seminar blog at lowrank.github.io || Reading (exposition) || Explains couplings and Monge plans, the manuscript's nonattainment claim, the supporting potential that identifies every optimizer, the mass-splitting obstruction and the equality of infima without an optimizing map, with proof outlines. || Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs.
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math, tables at repository commit <code>94c6630</code>, 28 October 2026) || Review (fidelity of the formal statements) || Rates this family "partial": the Lean statement covers the three-dimensional Coulomb density, the nonattainment by measure-preserving maps, the equality of the infima and the approximating maps, while the extension to every inverse-power exponent in each dimension at least two is not stated. It records a sandboxed Comparator pass for <code>CoulombCounterexample</code>. || MathVet states that it does not claim any theorem false and reports what was machine-checked as stated.
|-
| 2026-10-09 || [https://kokunoyumeto.github.io/open-math-courses-public/courses/three-marginal-coulomb-transport/ ''Three-marginal Coulomb transport''], a course on Open Mathematics Courses ([https://github.com/kokunoyumeto/open-math-courses-public kokunoyumeto/open-math-courses-public]) || Reading (exposition) || Four lessons: couplings and supporting potentials, two branches with a shared potential, Monge plans approaching the minimum, and a restatement with infinitesimals in which an internal pair of maps at an infinitely fine resolution exceeds the minimum by a positive infinitesimal. || The course states that it was written and self-checked by Claude Opus 5.5 (Anthropic) and that independent review is not complete.
|-
| 2026-10-10 || [https://github.com/carlostoledo1891/cert-machine/blob/a8f92f8e74893ff9afd167fd55310ad5afbc4958/corpus/openai-math/kernel-runs/38017370880/CoulombCounterexample.json cert-machine], openai/math lane K, run 38017370880 || Verification (two kernels) || Records that Lean's default kernel and the nanoda kernel accept the <code>CoulombCounterexample</code> solution, in 282 s. The project reports two-kernel certification of 398 of 416 challenges of the collection. || Repository.
|-
| 2026-10-10 to 2026-10-14 || [https://github.com/leanprover/con-leche/blob/90577d9e49ee3aed45da3d421bdf5dc780192304/DESIGN.md con-leche development journal] (task #348) || Verification (second kernel) || In its run over all Comparator challenges of the collection at commit <code>fd4aeeb</code>, the journal's table of exceptions does not list <code>CoulombCounterexample</code>, so it was accepted by con-leche and by Lean's official kernel. || Development journal.
|-
| 2026-10-16
| J. Jardine, Lattice24, referee review [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_545.json rank_545] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Reports no gap and no minor notes. The reviewer read the whole source and checked by hand the cost and gradient values at the central triples, the Hessian eigenvalues and their inverse-power analogues, the strict cost gaps for the other central patterns, the mass comparison in the splitting lemma, the marginals of the constructed optimal plan, the cell construction for the equality of infima and the Borel transport construction.
| Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0.
|-
| 2026-10-19 || ''AI Proved What?'', episode 373 [https://aiprovedwhat.com/episodes/373-coulomb-monge-counterexample/ Electrons that must flip a coin?], remade version of 24 October 2026 || Reading (exposition) || Video explainer of the claim that the optimal three-electron arrangement for one constructed density must split randomly between two families of triples. The episode states that real atoms and ground states are not claimed, that the strictly correlated energy is the same with or without maps, and that, as far as the site found, it is the first Coulomb example where no map at all is optimal. || Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 14 pages, dated September 25, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: the three-dimensional Coulomb theorem and the equality of the best values, with the extension to other powers not formalized. The page states that the paper does not cite an earlier, similarly titled counterexample for a different cost and that it found no specialist verdict.
|}

== Notes ==
* The background summarizes the manuscript's own account of earlier work.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]