A specific mathematical guess about random shapes stood for six years until an AI found the one shape that broke it.
A 2018 conjecture claimed that a certain type of symmetric shape always produced the smallest possible Gaussian minimum for random vectors. This belief shaped how mathematicians understood the density of random points in space. An AI-assisted optimization system searched the vast field of possibilities and identified a counterexample that human intuition had completely missed. The discovery forced a total revision of the theory and provided a new candidate for the actual minimum. This proves that even in pure mathematics, machine learning can see through logical blind spots that have stumped humans for years.
A revision of Litvak's conjecture on Gaussian minima and a volumetric zone conjecture
arXiv · 2605.02023
Litvak (2018) conjectured that, for any $p > 0$, the quantity $\mathbb{E}[\min_{i = 1}^n |g_i|^p]$ where $g \sim \mathcal{N}(0, \Sigma)$ is a centered Gaussian random vector is minimized among $n \times n$ correlation matrices $\Sigma$ by the Gram matrix of the regular simplex in $\mathbb{R}^{n - 1}$. We disprove this conjecture: the matrix with entries $\Sigma^{\mathrm{cos}}_{ij}=\cos(\pi(i - j) / n)$ already achieves a smaller moment for $p = 2$ and $n = 4$. We propose that $\Sigma^{\mathrm{co