Plectis Atlas
Search

↑ and ↓ move, Enter opens, Esc closes.

Show on the map

Nonattainment of the three-marginal Coulomb Monge problem

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

An optimal three-particle Coulomb configuration need not be a deterministic function of the first particle, even for smooth identical spatial densities. The Monge and Kantorovich infima agree, but the Monge infimum is not attained. The same phenomenon occurs for every inverse-power Riesz exponent in each dimension at least two, with a suitable density in each case. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments

What the papers state

The paper states

Theorem 1.1

In A counterexample to the Monge ansatz for the three-marginal Coulomb cost, 25 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

There is a nonnegative density \rho on \R^3 such that

\int_{\R^3}\rho(x)\dd x=1,\qquad \rho\in C_c^\infty(\R^3),\qquad \sqrt{\rho}\in C_c^\infty(\R^3),

with the following properties. For \mu=\rho\dd x, the value \E(\mu) is finite and is attained by a coupling in \Pi(\mu,\mu,\mu), but every pair of Borel maps T_2,T_3:\R^3\to\R^3 satisfying (T_2)_\#\mu=(T_3)_\#\mu=\mu obeys

\int_{\R^3}c\bigl(x,T_2(x),T_3(x)\bigr)\dd\mu(x)>\E(\mu).

The inequality permits the left-hand side to be infinite.

The release does not say which paper theorem its catalogue describes. The statement above shares the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.

Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
Papers and explanations
One current paper, readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

A counterexample to the Monge ansatz for the three-marginal Coulomb cost

25 September 2026, current edition.

Read it herePDF

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.

One Lean statement in this family

Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

CoulombCounterexample

The paper states

Theorem 1.1

In A counterexample to the Monge ansatz for the three-marginal Coulomb cost, 25 September 2026

There is a nonnegative density \rho on \R^3 such that

\int_{\R^3}\rho(x)\dd x=1,\qquad \rho\in C_c^\infty(\R^3),\qquad \sqrt{\rho}\in C_c^\infty(\R^3),

with the following properties. For \mu=\rho\dd x, the value \E(\mu) is finite and is attained by a coupling in \Pi(\mu,\mu,\mu), but every pair of Borel maps T_2,T_3:\R^3\to\R^3 satisfying (T_2)_\#\mu=(T_3)_\#\mu=\mu obeys

\int_{\R^3}c\bigl(x,T_2(x),T_3(x)\bigr)\dd\mu(x)>\E(\mu).

The inequality permits the left-hand side to be infinite.

The words they share: pi, coupling, rho, value, sqrt, attained, density, mu, finite, maps, three, monge, counterexample, coulomb, cost, borel, pair.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Coulomb Monge nonattainment with equality of infima

CoulombCounterexample.lean, the release lists it as a supporting statement.

theorem coulomb_counterexample_and_equal_infima :    ∃ rho : E3 → ℝ, HasFullCoulombConclusion rho := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).

Their note: coulomb_counterexample_and_equal_infima: there is a smooth compactly supported probability density ρ on ℝ³ (with smooth compactly supported √ρ) whose three-marginal Kantorovich problem for the cost Σ 1/|x_i-x_j| with identical marginals has finite value attained by some coupling, no pair of measurable measure-preserving maps (T₂,T₃) attains it, yet the Monge infimum equals the Kantorovich value and finite-cost Monge maps approach it. The summary’s extension to every inverse-power Riesz exponent in each dimension ≥ 2 is not stated (docs: ‘outside this statement’).

Definitions the group asks readers to compare with the paper: invDistance x y := (ofReal ‖x - y‖)⁻¹ in ℝ≥0∞ (value ⊤ on the diagonal), kantorovichValue/mongeValue as ⨅ over couplings / measure-preserving map pairs; all standard

“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).

The gap they name: The inverse-power (Riesz s > 0) extensions in every dimension d ≥ 2.

“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The Monge ansatz asks whether an optimal multi-marginal transport plan can be induced by maps from one marginal. For the three-marginal Coulomb cost in $\mathbb R^3$, the formalization constructs a smooth compactly supported probability density, with smooth compactly supported square root, for which no pair of measure-preserving Borel maps attains the Kantorovich minimum.

Nevertheless, the Monge and Kantorovich infima are equal: preserving maps with finite costs approach the minimum. This is the three-dimensional Coulomb result; the paper’s inverse-power extensions in every dimension are outside this statement.

AI-drafted comparisons, with no recorded human review

  • Coulomb Monge nonattainment with equality of infima CoulombCounterexample

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a smooth compactly supported density for which no pair of measure-preserving maps attains the three-marginal Coulomb minimum, with equality of the Monge and Kantorovich infima. The inverse-power extensions are outside it.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Coulomb Monge nonattainment with equality of infima CoulombCounterexample

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanationCompare the Lean statement with the paperSuggest a source attributionSuggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 373 No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 373 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form