# Multi-marginal transport representation audit

Family 373: Nonattainment of the three-marginal Coulomb Monge problem. First source-informed review, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Optimal-transport research teams can flag exact-attainment claims that restrict three-particle Coulomb couplings to two deterministic measure-preserving maps.

## Applicability and commercial boundary

The paper exhibits smooth compact densities where a Kantorovich optimum exists but the Monge representation does not attain it, even though the infima agree. The formal scope covers R3 Coulomb; broader Riesz extensions remain outside it. Approximating maps are not ruled out.

## Initial business decision

Conditional research. Commercial demand and profitability remain hypotheses. Research usefulness is not evidence of a buyer budget.

## Next verification action

Extract a discretized counterexample with an explicit relaxation gap or nonattainment indicator and distinguish numerical approximation from exact continuum attainment.

## Evidence scope

The catalog statement and main-paper abstract or introductory theorem passages were individually reviewed. Available family-level scope documents were inspected, with attention to version differences and exclusions. This is a first application triage, not a full proof audit or independent Lean verification. Family 376 has nine paper abstracts inspected; selected scopes are not assumed to cover all nine papers in full. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

## Pinned sources

- [A counterexample to the Monge ansatz for the three-marginal Coulomb cost](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/paper.pdf)
- [Selected formal scope](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/373.md)