Letrouit's square-root stability conjecture for Brenier maps from a uniform convex source

Let $\rho$ be the uniform probability measure on a compact convex body $K\subset\mathbb R^d$ , $d\ge2$ , and for a probability measure $\mu$ on a fixed compact set $Y$ let $T_\mu$ be the quadratic optimal (Brenier) map pushing $\rho$ to $\mu$ . Known target-uniform bounds had exponents such as $W_1^{1/6}$ (Delalande-Merigot) and $W_1^{2/15}$ ; Letrouit showed exponents above $1/3$ fail for some nonconvex sources and conjectured that for a uniform convex source $\|T_\mu-T_\nu\|_{L^2(\rho)}\le C\,W_2(\mu,\nu)^{1/2}$ uniformly over targets supported in $Y$ . Is this square-root estimate true?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Optimal transport; quantitative stability
Posed by: Cyril Letrouit (Unstable optimal transport maps, Conjecture 3)
Year posed: 2025
Years open: 1y
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: 12 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for $K$ compact convex with interior in $\mathbb R^d$ , $d\ge2$ , and $Y$ compact, $\|T_\mu-T_\nu\|_{L^2(\rho)}\le C(K,Y)W_2(\mu,\nu)^{1/3}$ for all probability measures $\mu,\nu$ on $Y$ , atomic or not; the exponent $1/3$ cannot be raised even for $K=Y=[-1,1]^d$ and three-atom targets, so Letrouit's square-root conjecture is false and $1/3$ is the sharp uniform exponent. Not shown: sharp exponents in $W_1$ (a sharp $W_1^{1/4}$ bound is attributed to Merigot) or for nonuniform source densities.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a 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 manuscripts are authored 'OpenAI' and name no human author. Single manuscript dated September 25, 2026.

Verification

No independent mathematician has checked this yet. Checked here: introduction and Theorem 1.1 read against Letrouit's Conjecture 3 as the manuscript cites it. Lean: the challenge lean/ComparatorChallenges/Brenier.json (solution_module OAI.Analysis.Brenier.Stability, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml. The statements one_third_stability and one_third_exponent_is_sharp were read here: a uniform constant for $W_2^{1/3}$ stability of unique quadratic optimal maps from the uniform measure on any compact convex body with interior, $d\ge2$ , and, for every exponent above $1/3$ and every constant, two three-atom targets on the cube violating the bound. The second disproves the conjecture; both state the headline. Not rebuilt here.

Letrouit's square-root stability conjecture for Brenier maps from a uniform convex source

What was actually shown

What the AI did

Verification

Sources

Discussion