What was already known
Colombo and Di Marino had proved equality of the Coulomb Kantorovich and cyclic-Monge infima for identical nonatomic marginals, without map attainment. Two-particle map existence and one-dimensional multi-particle map results did not settle the three-particle ℝ³ case.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Gives a smooth physical-density counterexample to attainment of the Coulomb co-motion ansatz.
Limitations
The checked statement concerns three marginals and the Coulomb cost in ℝ³. The manuscript’s inverse-power extensions in other dimensions are outside the Lean scope.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned solution covers smooth density, nonattainment by every preserving Borel pair, and equality of Monge and Kantorovich infima.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- There is a smooth compactly supported probability density on ℝ³, with smooth compactly supported square root, whose finite three-marginal Coulomb Kantorovich minimum is attained by a coupling but by no pair of measure-preserving Borel maps; nevertheless the Monge and Kantorovich infima are equal.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.
Read this published revision