Nonattainment of the three-marginal Coulomb Monge problem

Proves that the best three-electron Coulomb arrangement need not follow fixed maps, even for a smooth density.

1 paper September 25, 2026 Explained by AI

Chemists study how electrons arrange themselves when their repulsion dominates. One model asks for the cheapest way to place three electrons, each spread by the same density, under Coulomb repulsion. A common guess says the best arrangement fixes the second and third electron once the first is placed. This paper builds a smooth density where no such fixed rule is optimal. Fixed rules can still get as close to the best cost as you like.

First step

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Significance

Solid

It settles an open question about the co-motion ansatz in density functional theory, with a Lean statement of the three-dimensional Coulomb case.

Trust

Formal proof listed

Field

Partial differential equations

1 paper

The explanation

Start with three charged particles. The Coulomb cost of a triple is the sum of 1 over each pairwise distance. Put one particle at 0 and the others at e₁ and −e₁, one unit away on each side. The cost is 1 + 1 + 1/2 = 2.5. Now put them at 0, e₁ and e₂, two directions at right angles. The cost is 1 + 1 + 1/√2 ≈ 2.707. The first triple is cheaper. The triple 0, e₂, −e₂ costs 2.5 too.

Optimal transport here asks a question about a probability density, a rule for how mass spreads in space. Each of three electrons is spread by the same density. A plan is a joint rule for where all three sit at once. We want the plan with the least average cost. A Monge plan is a special plan: the first electron's position fixes the other two through maps. The guess that the best plan is a Monge plan is the Monge, or co-motion, ansatz. It comes from the strong-interaction limit of density functional theory, a method for computing electron behaviourSource 1.

For two electrons, Cotar, Friesecke and Klüppelberg proved an optimal map always existsSource 2. Colombo and Di Marino showed that maps get as close to the best cost as you likeSource 3. Friesecke still listed the three-electron question as open in January 2026Source 4.

The paper claims a smooth density where no pair of maps reaches the minimumSource 5. The density has five small blobs: one near 0 and one near each of ±e₁ and ±e₂. The triples (0, e_k, −e_k) have the same cost and the same pull at the centreSource 6. So every best plan uses two families of triples, one along e₁ and one along e₂. The centre blob holds mass 1/3 and each outer blob holds 1/6. Suppose a map picks the e₁ family on a set S of the centre. It sends mass ν(S)/3 into a region holding only ν(S)/6Source 7. That breaks the rule that each electron has the same density. So a best plan must split each point between the families, and no fixed map can. The idea adapts a splitting trick of Gerolin, Kausamo and RajalaSource 8.

Even so, the best cost over maps equals the true minimum; it is just never reachedSource 9. The paper extends this to inverse-power costs in every dimension at least 2Source 10.

How sure is this? Lean, a computer proof checker, has a formal statement of the three-dimensional Coulomb result. So the paper proves it, as far as the Lean statement matches the paper. The other powers and dimensions are not in the Lean scope. No specialist review of the paper is published.

What it doesn't show. This does not show that the guess fails for typical or physical densities from real atoms; it gives one constructed density. Lean covers only the three-dimensional Coulomb case.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

A solid counterexample with a Lean statement of the Coulomb case in three dimensions. It uses one constructed density, so it says nothing about typical physical densities. No specialist review is published.

Words used

Coulomb cost: For three points, the sum of 1 divided by each of the three pairwise distances.
Probability density: A rule for how a total mass of 1 spreads through space.
Optimal transport: Finding the cheapest way to couple distributions of mass under a given cost.
Marginal: The distribution of one particle alone, ignoring where the others are.
Monge plan: A plan in which the first particle's position fixes the others through maps.
Kantorovich minimum: The least cost over all plans, including ones that split mass between options.
Co-motion ansatz: The guess, from electron physics, that the best plan is a Monge plan.
Density functional theory: A method that computes electron behaviour from the electron density alone.

The original paper

A counterexample to the Monge ansatz for the three-marginal Coulomb cost (PDF, opens in a new tab)
September 25, 2026

Lean notes

All quoted lines (10)

Coulomb multi-marginal transport arises in the strong-interaction limit of density functional theory.
A counterexample to the Monge ansatz for the three-marginal Coulomb cost Back to text
For two particles, Cotar, Friesecke and Kl\"uppelberg proved existence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.
A counterexample to the Monge ansatz for the three-marginal Coulomb cost Back to text
Colombo and Di Marino proved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic maps for identical non-atomic marginals.
A counterexample to the Monge ansatz for the three-marginal Coulomb cost Back to text
The three-particle Coulomb question in $\R^3$ was still stated as open in January 2026 by Friesecke.
A counterexample to the Monge ansatz for the three-marginal Coulomb cost Back to text
We construct a smooth compactly supported probability density on $\R^3$ , 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.
A counterexample to the Monge ansatz for the three-marginal Coulomb cost Back to text
The triples $(0,e_k,-e_k)$ have the same Coulomb cost and the same central gradient.
A counterexample to the Monge ansatz for the three-marginal Coulomb cost Back to text
If it chooses a branch $H$ on a Borel set $S\subset B$ , it sends mass $\nu(S)/3$ into $H(S)$ , which has marginal mass only $\nu(S)/6$ .
A counterexample to the Monge ansatz for the three-marginal Coulomb cost Back to text
Our construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and Rajala for the repulsive harmonic cost $-\sum_{i<j}|x_i-x_j|^2$ .
A counterexample to the Monge ansatz for the three-marginal Coulomb cost Back to text
Nevertheless, the Monge and Kantorovich infima agree: we construct preserving maps whose costs approach the Kantorovich minimum.
A counterexample to the Monge ansatz for the three-marginal Coulomb cost Back to text
For every $d\ge2$ and $s>0$ , we also construct a smooth identical marginal with the same nonattainment and equality-of-infima properties for the inverse-power interaction $\sum_{i<j}|x_i-x_j|^{-s}$ on $\R^d$ .
A counterexample to the Monge ansatz for the three-marginal Coulomb cost Back to text

Other sources

openai/math, Lean scope note for family 373 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
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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