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Litvak's Conjecture on Gaussian Minima

Probability & statistics · posed by Alexander Litvak, 2018 · disproved

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Statement

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.

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  1. construction · #1

    AlphaEvolve, GPT-5.5 Pro, with Dmitriy Kunisky

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    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.

  2. Recorded elsewhere on #1 · not checked here

    recorded: correctVibeMathed site check

    scope Reproduction by the VibeMathed site

    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.

    Repeated from the source; nothing was checked here.

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