we-solve (beta)

Math-2466 · mathematics · Analysis

Monge Minimizers for Multimarginal Coulomb Transport

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Do Monge-type minimizers always exist for the multimarginal optimal-transport problem with Coulomb cost?

Formulation references: [S1], [S2].

Short English statement

Do Monge-type minimizers always exist for the multimarginal optimal-transport problem with Coulomb cost?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2466-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2466

Three-marginal Monge nonattainment: The smooth compactly supported three-dimensional density with three identical marginals has an attained finite Kantorovich minimum but no measure-preserving Borel Monge pair attaining it. A single such admissible Coulomb example refutes the catalogue universal Monge-minimizer assertion.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

Monge Minimizers for Multimarginal Coulomb Transport

OpenAI's claim

Three-marginal Monge nonattainment: The manuscript claims smooth compactly supported identical marginals in three dimensions for which no preserving pair of maps attains the Coulomb transport minimum. The Monge and relaxed infima nevertheless agree, and inverse-power analogues in other dimensions are also announced.

Mathematical statement of the claimed result

Mathematical claim 1.
For the three-marginal Coulomb cost $c(x_1,x_2,x_3)=\sum_{i<j}|x_i-x_j|^{-1}$, the claimed density $\rho\ge0$ has total mass one and $\rho,\sqrt\rho\in C_c^\infty(\mathbb R^3)$. Write $\mu=\rho\,dx$ and $E(\mu)=\inf_{\pi\in\Pi(\mu,\mu,\mu)}\int c\,d\pi$. This finite infimum is attained, but \[\int c(x,T_2x,T_3x)\,d\mu(x)>E(\mu)\] for every pair of Borel maps preserving $\mu$. Infinite cost on the left is allowed. Hence no Monge-type coupling attains the Kantorovich minimum even for this smooth compactly supported density.

Outline of the argument

Three-marginal Monge nonattainment: Two local families of triples share a supporting potential near a central component, while strict cost gaps exclude all other center patterns. Marginal mass imbalance then prevents a deterministic preserving map from selecting the required split between the two families. Approximate preserving maps approach the optimum, proving nonattainment without a gap between the infima.

Formal verification

Family 373 describes Lean coverage for the three-dimensional Coulomb nonattainment with equality of infima; the broader inverse-power extensions are not selected.

Three-marginal Monge nonattainment: The manuscript links CoulombCounterexample.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

Go to contribution →