Three-marginal Coulomb transport

OpenAI's counterexample (2026) to the Monge ansatz: for the Coulomb cost of three particles in space, and every Riesz cost in dimension at least two, a smooth density for which the optimal coupling exists but is given by no pair of maps, although maps come arbitrarily close. A fourth lesson restates the result with infinitesimals: an internal pair of maps at an infinitely fine resolution exceeds the minimum by a positive infinitesimal, and its standard part is the optimal coupling.

Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Couplings and supporting potentials
2. Two branches with a shared potential
3. Monge plans approach the minimum
4. Monge plans at infinitesimal resolution

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Inverse and implicit function theorems, change of variables, Tonelli's theorem and Borel probability measures.
• Zorn's lemma and dominated convergence, for the last lesson.