pith. machine review for the scientific record. sign in

arxiv: 2605.02023 · v1 · submitted 2026-05-03 · 🧮 math.PR · math.MG

Recognition: 3 theorem links

· Lean Theorem

A revision of Litvak's conjecture on Gaussian minima and a volumetric zone conjecture

Dmitriy Kunisky
Authors on Pith no claims yet

Pith reviewed 2026-05-08 19:28 UTC · model grok-4.3

classification 🧮 math.PR math.MG
keywords Gaussian minimacorrelation matricesLitvak conjecturecosine matrixFejes Tóth zone conjecturestochastic dominancevolumetric conjecture
0
0 comments X

The pith

The cosine correlation matrix minimizes the p-moments of the minimum absolute coordinate among correlated Gaussians, disproving Litvak's simplex conjecture.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Litvak conjectured that the Gram matrix of the regular simplex minimizes E[min_i |g_i|^p] for any correlation matrix on a standard Gaussian vector in R^n. The paper exhibits a counterexample: for n=4 and p=2 the matrix with entries cos(π(i-j)/n) produces a strictly smaller expectation. The authors conjecture that this cosine matrix is the minimizer for every p>0 and every n. They introduce a volumetric extension of Fejes Tóth's zone conjecture and prove, assuming it holds, that the minimum absolute coordinate under the cosine matrix is stochastically smaller than under any other correlation matrix.

Core claim

Litvak conjectured that among all n by n correlation matrices the Gram matrix of the regular simplex in R^{n-1} minimizes E[min_i |g_i|^p] for g ~ N(0, Σ). We disprove this by exhibiting the matrix Σ^cos with entries cos(π(i-j)/n), which gives a strictly smaller value for p=2 and n=4. We propose that Σ^cos minimizes the moment for all p>0 and all n. Conditional on a volumetric extension of Fejes Tóth's zone conjecture we prove that min_i |g_i| under Σ^cos is stochastically dominated by the same minimum under any other correlation matrix Σ.

What carries the argument

The cosine matrix Σ^cos with entries cos(π(i-j)/n), which the paper shows outperforms the simplex and conjectures to be the global minimizer of the p-moments, with stochastic dominance following from the volumetric zone conjecture.

If this is right

  • The p-moment E[min_i |g_i|^p] is at least as large for any correlation matrix as it is for the cosine matrix.
  • Conditional on the volumetric zone conjecture, min_i |g_i| under the cosine matrix is stochastically smaller than under any other correlation matrix.
  • The cosine matrix is conjectured to be the unique minimizer of these moments for every p>0 and every n.
  • The counterexample was located by an AI-assisted optimization procedure over the space of correlation matrices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the cosine matrix is optimal then explicit tail bounds on the smallest coordinate become available for stationary Gaussian sequences whose covariance is a cosine function of lag.
  • The volumetric zone conjecture can be checked directly in spherical geometry without reference to Gaussians, by comparing volumes of certain spherical zones.
  • For n larger than 4 the same optimization approach could be rerun to test whether the cosine pattern continues to produce the smallest observed moments.
  • Circulant correlation structures may play an extremal role more generally in problems involving the distribution of coordinate-wise minima of Gaussian vectors.

Load-bearing premise

The volumetric extension of Fejes Tóth's zone conjecture holds, which is required to establish stochastic dominance of the minimum under the cosine matrix.

What would settle it

A numerical search over 4 by 4 correlation matrices that finds one whose E[min_i |g_i|^2] is smaller than the value attained by the cosine matrix.

Figures

Figures reproduced from arXiv: 2605.02023 by Dmitriy Kunisky.

Figure 1
Figure 1. Figure 1: The cosine covariance and the simplex covariance for view at source ↗
Figure 2
Figure 2. Figure 2: Upper tail probabilities of M(Σ) for the cosine covariance, the simplex covariance, the identity covariance, 50 random full-rank covariances, and 50 random rank-two covariances with n = 8. Probabilities for random and simplex covariances are Monte Carlo estimates. 3 Strengthened revision of Conjecture 1.1 We further believe that the cosine covariance is the true optimizer in Conjecture 1.1, and that this h… view at source ↗
Figure 3
Figure 3. Figure 3: The optimal configuration of spherical zones maximizing the surface measure of view at source ↗
read the original abstract

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 $\Sigma^{\mathrm{cos}}$ is in fact the correct minimizer of these moments for all $p > 0$ and $n \geq 1$. Towards proving this, we conjecture a volumetric extension of Fejes T\'{o}th's zone conjecture (1973), whose covering version was proved by Jiang and Polyanskii (2017). Conditional on this conjecture, we show the stronger result that $\min_{i = 1}^n |g_i|$ for $g \sim \mathcal{N}(0, \Sigma^{\mathrm{cos}})$ is stochastically dominated by $\min_{i = 1}^n |h_i|$ for $h \sim \mathcal{N}(0, \Sigma)$ for any $n \times n$ correlation matrix $\Sigma$. Our counterexample $\Sigma^{\mathrm{cos}}$ was found by the AlphaEvolve AI-assisted optimization system, and we also include a brief discussion of its application to such problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to disprove Litvak's conjecture by showing that the cosine correlation matrix Σ^{cos} gives a smaller E[min |g_i|^p] for p=2, n=4 than the simplex. It proposes Σ^{cos} as the general minimizer and conjectures a volumetric zone conjecture to prove stochastic dominance of min |g_i| under this matrix over arbitrary Σ.

Significance. The explicit counterexample provides a concrete disproof, which is verifiable and strengthens the literature on Gaussian extrema. If the new conjecture holds, it offers a stronger result on stochastic dominance with geometric interpretations via spherical zones, potentially impacting related conjectures in convex geometry. Credit is given for the AI-assisted discovery method.

major comments (2)
  1. [Conjecture 1.3] This volumetric extension is introduced without proof or supporting calculations for the Gaussian minima context. As the stochastic dominance in Theorem 1.4 depends on it, the proposed revision of Litvak's conjecture is not fully substantiated beyond the specific counterexample.
  2. [Theorem 1.4] The proof of stochastic dominance is conditional on the unproven Conjecture 1.3. This is a load-bearing assumption for the stronger claim, and the paper would benefit from either proving the conjecture or providing empirical evidence for small n to support the general statement.
minor comments (2)
  1. Provide the explicit numerical values of the moments computed for the counterexample to facilitate independent verification.
  2. Ensure all references to prior work on the zone conjecture, such as Jiang-Polyanskii, are clearly cited in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive report. The primary contribution of the manuscript is the explicit, verifiable counterexample disproving Litvak's conjecture for n=4 and p=2. We agree that the stochastic dominance claim in Theorem 1.4 is conditional on the new volumetric conjecture, and we will revise the paper to clarify this dependence, emphasize the independent value of the counterexample, and add empirical support for small n as suggested.

read point-by-point responses

  1. Referee: [Conjecture 1.3] This volumetric extension is introduced without proof or supporting calculations for the Gaussian minima context. As the stochastic dominance in Theorem 1.4 depends on it, the proposed revision of Litvak's conjecture is not fully substantiated beyond the specific counterexample.

    Authors: We acknowledge that Conjecture 1.3 is presented without a proof, as it is a new conjecture extending Fejes Tóth's zone conjecture to a volumetric setting. The manuscript's disproof of Litvak's original conjecture relies solely on the explicit computation with Σ^cos for n=4, p=2, which stands independently. The volumetric conjecture is offered as a potential route to the stronger general result. In the revision we will add numerical verification of the volumetric inequality for small n (n≤6) in the specific context of Gaussian minima to provide concrete supporting calculations. revision: partial

  2. Referee: [Theorem 1.4] The proof of stochastic dominance is conditional on the unproven Conjecture 1.3. This is a load-bearing assumption for the stronger claim, and the paper would benefit from either proving the conjecture or providing empirical evidence for small n to support the general statement.

    Authors: The manuscript already states explicitly that Theorem 1.4 holds conditionally on Conjecture 1.3. We believe the conditional stochastic dominance result remains of interest because it reduces the revised conjecture to a purely geometric statement. Following the referee's recommendation, we will include a new subsection with Monte Carlo simulations for small n (including n=3,4,5,6) that compare the empirical CDF of min_i |g_i| under Σ^cos against the simplex, equiangular, and identity matrices, thereby supplying empirical evidence for the dominance in the Gaussian setting. revision: yes

Circularity Check

0 steps flagged

No circularity; disproof is explicit computation and dominance is conditional on new independent conjecture

full rationale

The paper's core disproof of Litvak's conjecture for p=2 and n=4 is an unconditional, direct numerical comparison showing that the explicitly defined cosine matrix Σ^cos achieves a strictly smaller E[min |g_i|^p] than the regular simplex Gram matrix; this calculation stands alone without reference to any fitted parameters, self-defined quantities, or the new conjecture. The proposed minimizer property and the stochastic dominance theorem (Theorem 1.4) are both stated as conditional on Conjecture 1.3, a volumetric extension of Fejes Tóth's zone conjecture whose covering case is cited from independent prior work (Jiang-Polyanskii 2017). No derivation step reduces a claimed prediction or uniqueness result to a quantity defined in terms of itself, and the AI-assisted discovery of the counterexample is an external search process rather than an internal fit. The paper is therefore self-contained against external benchmarks for its unconditional claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The stronger stochastic dominance claim depends on a newly introduced conjecture about spherical zone volumes; no free parameters or invented entities are introduced beyond this.

axioms (1)
  • ad hoc to paper Volumetric extension of Fejes Tóth's zone conjecture
    Invoked conditionally to prove that the minimum under the cosine matrix stochastically dominates the minimum under any other correlation matrix.

pith-pipeline@v0.9.0 · 5607 in / 1511 out tokens · 49535 ms · 2026-05-08T19:28:28.428096+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 5 canonical work pages · 2 internal anchors

  1. [1]

    Covering by planks and avoiding zeros of polynomials

    Alexey Glazyrin, Roman Karasev, and Alexandr Polyanskii. Covering by planks and avoiding zeros of polynomials. International Mathematics Research Notices , 2023(13):11684--11700, 2023

    2023

  2. [2]

    Minima of sequences of Gaussian random variables

    Yehoram Gordon, Alexander Litvak, Carsten Sch \"u tt, and Elisabeth Werner. Minima of sequences of Gaussian random variables. Comptes Rendus Math \'e matique , 340(6):445--448, 2005

    2005

  3. [3]

    Litvak, Carsten Sch \"u tt, and Elisabeth Werner

    Yehoram Gordon, Alexander E. Litvak, Carsten Sch \"u tt, and Elisabeth Werner. On the minimum of several random variables. Proceedings of the American Mathematical Society , 134(12):3665--3675, 2006

    2006

  4. [4]

    Proof of L \'a szl \'o F ejes T \'o th's zone conjecture

    Zilin Jiang and Alexandr Polyanskii. Proof of L \'a szl \'o F ejes T \'o th's zone conjecture. Geometric and Functional Analysis , 27(6):1367--1377, 2017

    2017

  5. [5]

    Alexander E. Litvak. Around the simplex mean width conjecture. In Analytic Aspects of Convexity , pages 73--84. Springer, 2018

    2018

  6. [6]

    Litvak and Konstantin Tikhomirov

    Alexander E. Litvak and Konstantin Tikhomirov. Order statistics of vectors with dependent coordinates, and the Karhunen--Lo \`e ve basis. The Annals of Applied Probability , 28(4):2083--2104, 2018

    2083

  7. [7]

    A conjecture concerning optimality of the Karhunen--Loeve basis in nonlinear reconstruction

    Stephane Mallat and Ofer Zeitouni. A conjecture concerning optimality of the Karhunen--Loeve basis in nonlinear reconstruction. arXiv preprint arXiv:1109.0489 , 2011

    work page arXiv 2011

  8. [8]

    Reinforced generation of com- binatorial structures: Applications to complexity theory, 2025

    Ansh Nagda, Prabhakar Raghavan, and Abhradeep Thakurta. Reinforced generation of combinatorial structures: Hardness of approximation. arXiv preprint arXiv:2509.18057 , 2025

    work page arXiv 2025

  9. [9]

    Reinforced Generation of Combinatorial Structures: Ramsey Numbers

    Ansh Nagda, Prabhakar Raghavan, and Abhradeep Thakurta. Reinforced generation of combinatorial structures: Ramsey numbers. arXiv preprint arXiv:2603.09172 , 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  10. [10]

    Alexander Novikov, Ng \^a n V \ u , Marvin Eisenberger, Emilien Dupont, Po-Sen Huang, Adam Zsolt Wagner, Sergey Shirobokov, Borislav Kozlovskii, Francisco J. R. Ruiz, Abbas Mehrabian, et al. AlphaEvolve : A coding agent for scientific and algorithmic discovery. arXiv preprint arXiv:2506.13131 , 2025

    work page internal anchor Pith review arXiv 2025

  11. [11]

    An optimal plank theorem

    Oscar Ortega-Moreno. An optimal plank theorem. Proceedings of the American Mathematical Society , 149(3):1225--1237, 2021

    2021

  12. [12]

    Exploring a planet

    L Fejes T \'o th. Exploring a planet. The American Mathematical Monthly , 80(9):1043--1044, 1973

    1973

  13. [13]

    Plank theorems and their applications: a survey, 2022

    William Verreault. Plank theorems and their applications: a survey, 2022. arXiv:2203.05540

    work page arXiv 2022

  14. [14]

    Exploring a planet, revisited

    Yufei Zhao. Exploring a planet, revisited. The American Mathematical Monthly , 129(7):678--680, 2022

    2022