VibeMathedMath problems solved by AI
Sign in

Litvak's Conjecture on Gaussian Minima

Litvak conjectured in 2018 that for every p>0p > 0 the quantity E[min⁡i≤n∣gi∣p]\mathbb{E}[\min_{i \le n} |g_i|^p], for g∼N(0,Σ)g \sim \mathcal{N}(0,\Sigma), is minimized over n×nn \times n correlation matrices by the Gram matrix of the regular simplex in Rn−1\mathbb{R}^{n-1}. False: the matrix Σijcos⁡=cos⁡(π(i−j)/n)\Sigma^{\cos}_{ij} = \cos(\pi(i-j)/n) already gives a strictly smaller value at p=2p = 2, n=4n = 4.

Result
Disproved(see note)
Status
Resolved
AI contribution
AI-discovered
Method
Construction
Field
High-dimensional probability
Posed by
Alexander Litvak
Year posed
2018
Years open
8y
Solved
2026-05-03
Model
AlphaEvolve, GPT-5.5 Pro
Vendor
—
Collaborators
Dmitriy Kunisky
Verification
Site-confirmed
Publication
Preprint
Significance
20 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

the paper proposes that the cosine matrix is the true minimizer for all p and n, and proves a stronger stochastic domination statement conditional on a new volumetric extension of Fejes Toth's zone conjecture

What the AI did

Two separate model contributions, and the author keeps them apart. The counterexample itself came out of black-box minimization with AlphaEvolve; what made it usable was the author then recognising a continuous function underneath the numbers and identifying it as the cosine, which turned a numerical optimum into a clean matrix and then into a stronger conjecture. He notes conventional optimizers such as differential evolution and BFGS also produced matrices that would disprove the conjecture, but rarely converged to this one. Separately, GPT-5.5 Pro surfaced the connection to Fejes Toth's zone conjecture, which the author had not known about, while repeatedly producing proofs that leaned on an unsupported step as though it were established. Isolating that step is what produced the volumetric conjecture stated in the paper, so the model's mistake was itself informative.

Verification

Reproduced here. For Σcos⁡\Sigma^{\cos} with n=4n = 4 the matrix has rank two, so gi=Rcos⁡(Θ−πi/4)g_i = R\cos(\Theta - \pi i/4) with R2∼χ22R^2 \sim \chi^2_2, giving the closed form E[min⁡i∣gi∣2]=1−22/π=0.0996836838\mathbb{E}[\min_i |g_i|^2] = 1 - 2\sqrt{2}/\pi = 0.0996836838. The regular-simplex Gram matrix was evaluated by deterministic quadrature over the sphere, stable at 0.14218330.1421833 across grid refinements and corroborated by a 20-million-sample simulation at 0.1422380.142238. The cosine matrix is smaller by about 30 percent, far outside any numerical doubt. arXiv preprint, not peer-reviewed.

Source

arXiv:2605.02023 - A revision of Litvak's conjecture on Gaussian minima and a volumetric zone conjecture

Submitted by Curator34

Discussion