Nonattainment of the three-marginal Coulomb Monge problem

Source and verification

This article explains family 373 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

In optimal transport, allowing randomized couplings can produce an optimizer even when no deterministic map attains the same value. Family 373 claims this phenomenon for three repelling particles with the same smooth spatial density. Equality of infima survives; attainment by maps fails.

🏷️ Preliminaries: couplings and Monge plans

Let $\mu$ be a probability measure on ${\mathbb{R}}^3$. A coupling $\pi \in \Pi (\mu ,\mu ,\mu )$ is a joint distribution of three positions with each marginal equal to $\mu$. Its Coulomb cost is

$$c(x_1,x_2,x_3)=\sum _{1\le i<j\le 3}\frac{1}{|x_i-x_j|},$$

with infinite cost when positions coincide. The Kantorovich problem minimizes $\int cd\pi$ over all couplings.

A Monge plan has the special form $(\mathrm{Id},T_2,T_3{)}_{\#}\mu$, where each Borel map preserves $\mu$. Once the first position is known, the other two are determined. A general coupling may instead split its conditional mass among several configurations.

The distinction is not whether one can find an optimizer that is not a graph. Even if such an optimizer exists, another optimizer might still be induced by maps. To refute the Monge ansatz, every preserving pair of maps must be excluded.

🏷️ The manuscript’s nonattainment claim

Family 373, as stated in the manuscript

There is a smooth compactly supported probability density $\rho$ on ${\mathbb{R}}^3$, with smooth compactly supported square root, for which the three-marginal Coulomb minimum is finite and attained by a coupling, but not by any pair of measure-preserving Borel maps. Nevertheless, the Monge and Kantorovich infima are equal. (OpenAI, 2026)

The same manuscript extends its construction to every inverse-power cost ${\sum }_{i<j}|x_i-x_j{|}^{-s}$, $s>0$, in each dimension at least two, with a suitable density chosen for each case. The formalization scope described below selects the three-dimensional Coulomb statement.

🏷️ A supporting potential identifies every optimizer

Suppose a function $h$ satisfies

$$\begin{array}{c}c(x_1,x_2,x_3)\\ \ge h(x_1)+h(x_2)+h(x_3)\end{array}$$

on the support of the marginals. Integrating gives $\int cd\pi \ge 3\int hd\mu$ for every coupling. If one coupling attains equality, it is optimal. Moreover, any optimizer must be supported where the pointwise inequality is an equality, since the nonnegative deficit has integral zero.

The manuscript constructs two families of equality triples,

$$(x,Y_k(x),W_k(x)),k=1,2,$$

with $x$ in a small central ball and the other points in four disjoint outer components near $\pm e_1,\pm e_2$. Local calculus produces a shared supporting potential. Strict cost gaps exclude other configurations, while Coulomb repulsion excludes repeated components after their neighborhoods are made small enough.

This supporting inequality is the geometric part of the proposed proof. It constrains all optimizers, including those generated by highly irregular maps.

🏷️ The mass-splitting obstruction

Give the central component measure $\nu /3$, where $\nu$ is a smooth probability measure, and give each outer component the appropriate pushforward of $\nu /6$. Choosing either branch with probability one half and then randomly permuting the three positions produces the desired equal marginals and saturates the supporting inequality.

Now suppose a deterministic optimizer exists. On the central component, its second coordinate must choose one of the four allowed outer images. For any one branch $H$, let $S$ be the set on which it chooses $H$. The map sends mass $\nu (S)/3$ into $H(S)$, but the target marginal assigns that same image only $\nu (S)/6$. Preservation is impossible if $\nu (S)>0$.

There are only four choices, and together they cover the central component almost everywhere. At least one selection set must have positive measure. This contradicts preservation and excludes every Borel selection, not merely continuous or cyclic maps.

🏷️ Equal infima without an optimizing map

The source also constructs preserving maps with costs approaching the optimal value. Thus there is no fixed positive gap between the two infima. Fine deterministic assignments can approximate the mass splitting weakly, while their limit is a randomized coupling rather than a graph.

This is a loss of compactness of the class of deterministic plans under the convergence relevant to transport. The compact relaxed problem attains its minimum; the smaller graph class can approach it without containing it. Smoothness of the marginal does not repair that structural obstruction.

🏷️ Key steps and proof outlines

Building a contact surface by strong convexity

The local construction begins at $(0,a,-a)$ with $a=e_1$ or $e_2$. Both triples have cost $5/2$ and zero derivative in the central variable. This permits the same central potential

$$u_0(x)=\frac{5}{2}-\frac{K}{2}|x{|}^2$$

for both branches. At the negative outer component use $v_a(z)=\frac{5}{4}a\cdot (z+a)-\frac{K}{2}|z+a{|}^2$. Hold the positive outer point $y$ fixed and minimize

$$F_a(x,y,z)=c(x,y,z)-u_0(x)-v_a(z)$$

over $(x,z)$ near $(0,-a)$.

Proof outline. At the center triple, the $(x,z)$ Hessian is $M+K{I}_6$, where $M=D_{(x,z)}^{2}c(0,a,-a)$. Large $K$ makes it positive definite, and continuity preserves this on a sufficiently small neighborhood. The implicit function theorem gives a unique minimizing pair $(X_a(y),Z_a(y))$, and its minimum value defines the missing potential $w_a(y)$. Strong convexity gives both the supporting inequality and its exact equality set.

The additional issue is invertibility of each coordinate map. For $h(b)=|b{|}^{-1}$,

$$D^{2}h(b)=|b{|}^{-3}\left(3\frac{b{b}^T}{|b{|}^2}-I\right)$$

has eigenvalues $2|b{|}^{-3},-|b{|}^{-3},-|b{|}^{-3}$. The two blocks of the mixed derivative $N=D_y{\nabla }_{(x,z)}c(0,a,-a)$ are therefore invertible. Since

$$\begin{array}{c}D(X_a,Z_a)(a)=-(M+K{I}_6{)}^{-1}N,\\ K(M+K{I}_6{)}^{-1}N⟶N,\end{array}$$

both coordinate derivatives at $y=a$ are invertible for sufficiently large $K$, and continuity preserves their invertibility locally. Thus the central coordinate can be inverted and used as the common parameter for the two contact branches. (OpenAI, 2026)

Why unwanted triples cannot also be optimal

Local minimization only certifies triples in the two intended component patterns. The source extends this to all five components by a finite gap argument. At the component centers the potential values are $5/2$ at zero and zero at the four outer points. A triple containing zero and points from different opposite pairs has cost

$$2+\frac{1}{\sqrt{2}}>\frac{5}{2}.$$

A triple of distinct outer centers also has strictly positive cost while its potential sum is zero. These finitely many strict gaps persist when the neighborhoods shrink. Finally, if two points occupy the same radius-$r$ component, their pair cost is at least $(2r{)}^{-1}$. This exceeds every possible potential sum once $r$ is small, because the potentials are bounded on the preliminary neighborhoods.

The result is a certificate whose equality set consists only of the two prescribed families and their permutations. The measure-selection contradiction above depends on this exhaustiveness: constructing one split optimizer would not exclude a different graph optimizer. (OpenAI, 2026)

Approximating the optimizer without approaching collisions

Let $K_0,\dots ,K_4$ be the compact supports inside the five components and let $\eta >0$ be their minimum pairwise distance. Partition them into cells $A_i$ of diameter at most $\delta$. For the constructed optimal coupling ${\pi }_0$, record

$$q_{ijk}={\pi }_0(A_i\times A_j\times A_k).$$

Proof outline. Split each source cell $A_i$ into Borel pieces of masses $q_{ijk}$; absolute continuity permits these splits by coordinate slabs. On each positive-mass piece, choose Borel maps with output laws equal to the normalized restrictions of $\mu$ to $A_j$ and $A_k$. Conditional distribution functions and inverse quantiles provide such maps. Summing the output weights gives exactly the original target marginal, because ${\sum }_{i,k}{q}_{ijk}=\mu (A_j)$ and similarly for the third coordinate.

The resulting graph plan has the same mass in every cell product as ${\pi }_0$. Every active product uses three different components, so all its pairwise distances remain at least $\eta$. Moving each coordinate by at most $\delta$ changes each reciprocal distance by at most $2\delta /{\eta }^2$. Consequently

$$\left|\int cd{\pi }_{\delta }-\int cd{\pi }_0\right|\le \frac{6\delta }{{\eta }^2}.$$

This quantitative estimate proves equality of infima despite strict nonattainment. Uniform separation is essential: weak convergence alone would not justify passing the singular Coulomb cost to the limit. (OpenAI, 2026)

See Also

• Family 374 treats quadratic two-marginal transport, where an absolutely continuous source does determine a unique optimal map.
• Family 368 concerns approximation while retaining an additional geometric constraint on maps.

🏷️ Manuscripts and verification scope

• A counterexample to the Monge ansatz for the three-marginal Coulomb cost — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope page selects smooth three-dimensional Coulomb Monge nonattainment with equality of infima. The inverse-power extensions in other dimensions are outside that statement. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. A counterexample to the Monge ansatz for the three-marginal Coulomb cost.