# Sharp one-third stability of Brenier maps > Proves optimal transport maps from a uniform convex source change at most like the cube root of the target change. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-374/ Last updated: 2026-10-14 **Proves optimal transport maps from a uniform convex source change at most like the cube root of the target change.** **Trust:** Formal proof listed · **Papers:** 1 · **First paper:** September 25, 2026 · **Field:** Partial differential equations · **Entry:** 374 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. Optimal transport asks for the cheapest way to move a pile of sand into a new shape. The best plan is a map that tells each grain where to go. If the target shape changes a little, how much can that map change? The paper shows the change is at most a constant times the cube root of the target change. It also shows the cube root cannot be improved, which disproves a conjectured square root. ## At a glance - **First step:** — - **Significance:** Solid. It finds the exact exponent for a well-studied stability question and disproves a recent conjecture, with a Lean statement of both parts. - **Trust:** Formal proof listed - **Field:** Partial differential equations (1 paper) ## The explanation Start with sand spread evenly over the square from −1 to 1 in both directions. We send it to three points. Sand with first coordinate below −a goes to (−1, 0). Sand above a goes to (1, 0). The middle strip, of width 2a, goes to (0, 0). Now tilt the strip slightly. Its edges move to |x₁| < a + (a/2)·x₂, and the middle point moves to (0, a/2). The strip keeps the same amount of sand. How far apart are the two targets? The Wasserstein distance W₂ measures the cheapest way to move one target into the other. Here only the middle point moves, by a/2, carrying mass a. So W₂ = a^{3/2}/2. For a = 0.01 this is 0.0005. How far apart are the two maps? Some sand near the strip edges switches destination by about 1. The average squared gap gives an L² distance of about 0.0500. Shrink a to 0.001. Then W₂ ≈ 0.0000158 and the map gap is ≈ 0.0158. Divide the map gap by W₂ raised to a power. With power 1/2 the ratio grows: 2.24, then 3.98. With power 1/3 it stays near 0.63. So a square-root bound cannot hold, but a cube-root bound might. The best map for squared distance is called a Brenier map. A Hölder bound with exponent 1/3 says the map gap is at most C·W₂^{1/3}. Earlier bounds had weaker exponents, such as 2/15 and 1/6, in a related distance W₁[1,2]. One sharper bound had a constant that depended on the number and masses of the points[3]. Letrouit conjectured a square-root bound for uniform convex sources[4]. The paper claims the cube-root bound for the uniform measure on any compact convex body, in every dimension d ≥ 2[5]. The constant does not depend on the number of points, their masses, or their spacing[6]. The body need not be smooth[7]. The tilted-strip family above, with three target points, shows no larger exponent works. This disproves the square-root conjecture[8,9]. The new step is a linear bound, in W₂, on how much the potential function behind the map changes[10]. How sure is this? Lean, a computer proof checker, has formal statements of both the cube-root bound and its sharpness. So the paper proves them, as far as the Lean statements match the paper. No specialist review of the paper is published. **What it doesn't show.** This does not cover sources that are not uniform on a convex body. It does not give the best constant, and it does not improve bounds stated with the W₁ distance. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check A solid, sharp result with Lean statements of the bound and its sharpness. It covers only uniform sources on convex bodies. No specialist review is published. ## Words used - **Optimal transport:** Finding the cheapest way to move one distribution of mass into another. - **Brenier map:** The optimal transport map when cost is the squared distance moved. - **Wasserstein distance W₂:** The square root of the least average squared distance needed to turn one distribution into another. - **L² distance:** The square root of the average squared gap between two maps. - **Hölder bound:** A bound of the form C times a distance raised to a fixed power, here 1/3. - **Convex body:** A bounded closed shape with interior that contains the segment between any two of its points. - **Sharp exponent:** A power that works, while every larger power fails. ## The original paper - [Sharp One-Third Stability of Brenier Maps](https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf) - September 25, 2026 - [Lean notes](https://github.com/openai/math/blob/main/lean/docs/374.md) ## Quoted lines from the paper (10) [1] From [Sharp One-Third Stability of Brenier Maps](https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf): > M\'erigot, Delalande, and Chazal used dualization to obtain target-uniform bounds from Berman's work, with dimension-dependent exponents, and then proved a dimension-independent \(W_1^{2/15}\) bound for a uniform convex source. [2] From [Sharp One-Third Stability of Brenier Maps](https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf): > Delalande and M\'erigot proved an \(L^2\)-map bound of order \(W_1^{1/6}\) for source densities bounded above and below on a convex domain and targets in a fixed compact set. [3] From [Sharp One-Third Stability of Brenier Maps](https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf): > Its constant depends on the number of atoms, a positive lower bound for their masses, and separation and angular geometry, whereas … has one constant for all Borel targets, including degenerating atomic measures. [4] From [Sharp One-Third Stability of Brenier Maps](https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf): > Letrouit proved an obstruction to exponents above \(1/3\) for a nonconvex source and formulated uniform square-root stability in \(W_2\) for a uniform convex source as a conjecture. [5] From [Sharp One-Third Stability of Brenier Maps](https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf): > For the uniform probability measure \(\rho\) on a compact convex body in \(\R^d\), \(d\ge2\), quadratic optimal transport maps are one-third H\"older continuous in \(L^2(\rho)\) with respect to the target's \(2\)-Wasserstein distance. [6] From [Sharp One-Third Stability of Brenier Maps](https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf): > The constant is independent of atom count, positive atom masses, site separation, and density bounds. [7] From [Sharp One-Third Stability of Brenier Maps](https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf): > Neither smoothness nor strict convexity of \(K\) is required, and \(Y\) need not be convex. [8] From [Sharp One-Third Stability of Brenier Maps](https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf): > A three-atom family on a fixed cube disproves Letrouit's conjectured uniform square-root estimate. [9] From [Sharp One-Third Stability of Brenier Maps](https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf): > Tilting the middle affine function preserves its cell mass but switches source points between slopes at order-one separation. [10] From [Sharp One-Third Stability of Brenier Maps](https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf): > Their interpolation inequality is the antecedent of our last step; the improvement here is the linear \(W_2\) estimate for potential differences. ## Other sources - [openai/math, Lean scope note for family 374](https://github.com/openai/math/blob/main/lean/docs/374.md): “The formalization proves one-third Hölder stability of Brenier maps from the uniform measure on a compact convex body with nonempty interior” - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” ## Related - [Nonattainment of the three-marginal Coulomb Monge problem](https://closedform.nishachay.tech/e/openai-math-2026-373/) - Partial differential equations - [De Giorgi's conjecture in dimension eight](https://closedform.nishachay.tech/e/openai-math-2026-375/) - Partial differential equations - [Global uniqueness in smooth isotropic elasticity](https://closedform.nishachay.tech/e/openai-math-2026-372/) - Partial differential equations - [Universal computation in forced Navier–Stokes flows](https://closedform.nishachay.tech/e/openai-math-2026-376/) - Partial differential equations - [Stable blowup for the defocusing Schrödinger equation](https://closedform.nishachay.tech/e/openai-math-2026-371/) - Partial differential equations - [Partial differential equations](https://closedform.nishachay.tech/science/mathematics/partial-differential-equations/) - Field · 16 results - [Lean-checked results](https://closedform.nishachay.tech/trust/lean-checked/) - Trust · 242 results - [Solid results](https://closedform.nishachay.tech/significance/solid/) - Significance · 50 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20374%5D%20)