Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 373-01. A counterexample to the Monge ansatz for the three-marginal Coulomb cost
Manuscript map · lines 9505–9514
OpenAI@2026-10-07#rem:oai-p-373-01
Authors: OpenAI. Family 373.
Title: A counterexample to the Monge ansatz for the three-marginal Coulomb cost.
Abstract: We construct a smooth compactly supported probability density on ℝ3, 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. Nevertheless, the Monge and Kantorovich infima agree: we construct preserving maps whose costs approach the Kantorovich minimum. For every d ≥ 2 and s > 0, we also construct a smooth identical marginal with the same nonattainment and equality-of-infima properties for the inverse-power interaction ${}sum_{i{}lt j}|x_i-x_j|{}{-s}$ on ℝd.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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