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The Monge (co-motion) ansatz for three-marginal Coulomb transport in R3\mathbb R^3: is the optimum always attained by maps?

For a probability measure μ\mu on R3\mathbb R^3, minimize ∫c dπ\int c\,d\pi over couplings π\pi with three marginals equal to μ\mu, where c(x1,x2,x3)=∑i<j∣xi−xj∣−1c(x_1,x_2,x_3)=\sum_{i<j}|x_i-x_j|^{-1}. In the strictly correlated electron limit of density functional theory one expects optimizers of co-motion (Monge) form (Id,T2,T3)#μ(\mathrm{Id},T_2,T_3)_\#\mu with measure-preserving maps. This holds for two marginals (Cotar-Friesecke-Kluppelberg) and in one dimension (Colombo-De Pascale-Di Marino). If μ\mu is absolutely continuous with finite Coulomb transport value, is the minimum always attained by a Monge plan?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Multi-marginal optimal transport; density functional theory
Posed by
Recorded by G. Friesecke, A. Gerolin and P. Gori-Giorgi (2022 preprint of their DFT chapter, Theorem 2.4 and discussion); still stated open by Friesecke (January 2026)
Year posed
2022
Years open
4y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
22 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there is ρ≥0\rho\ge0 with ∫ρ=1\int\rho=1 and ρ,ρ∈Cc∞(R3)\rho,\sqrt\rho\in C_c^\infty(\mathbb R^3) such that for μ=ρ dx\mu=\rho\,dx the three-marginal Coulomb minimum is finite and attained, but every pair of Borel maps T2,T3T_2,T_3 pushing μ\mu to itself has cost strictly above the minimum. The Monge and Kantorovich infima are nevertheless equal. The same holds for every Riesz cost ∑i<j∣xi−xj∣−s\sum_{i<j}|x_i-x_j|^{-s}, s>0s>0, in every dimension d≥2d\ge2, with a suitable density. It concerns one constructed density: it does not say Monge plans fail generically, and no external potential realizing the density is asserted.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against the question as Friesecke states it. lean/docs/373.md points to ComparatorChallenges/CoulombCounterexample.json (theorem OAI.Problem356.coulomb_counterexample_and_equal_infima, solution module OAI.Analysis.CoulombTransport.Main); the solution file exists at the pinned commit and the challenge is not in formalization.yaml. The statement was read here: there is a smooth compactly supported probability density on R3\mathbb R^3 with smooth compactly supported square root whose three-marginal Coulomb Kantorovich value is finite and attained, every pair of measure-preserving measurable maps has strictly larger cost, and the Monge infimum equals the Kantorovich value. That is the headline claim. Not rebuilt here. The inverse-power Riesz extensions are outside the Lean statement.

Sources

Changelog1 change

Discussion