Talagrand’s convolution conjecture

On the Boolean hypercube $G = \{-1,1\}^n$ with uniform measure $\lambda$ , let $T_\mu f(x) = \int_G f(x \odot y)\,d\mu(y)$ be convolution by a finite positive measure $\mu$ , and set $\psi_\mu(u) = \sup\{u\,\lambda(\{T_\mu f \ge u\}) : f \ge 0,\ \|f\|_1 = 1\},$ which measures how much better than Markov's inequality convolution makes the tail. In 1989 Talagrand conjectured that for the biased-coin product measure $\mu_a = (\tfrac{1+a}{2}\delta_1 + \tfrac{1-a}{2}\delta_{-1})^{\otimes n}$ with $0 < a < 1$ , $\psi_{\mu_a}(u) \le \frac{C_a}{\sqrt{\log u}} \qquad (u > 1),$ with $C_a$ depending on $a$ alone and not on the dimension $n$ . He offered a \$1000 prize for a proof. The Gaussian analogue was settled by Eldan and Lee; the hypercube case, the original, stayed open.

This paper claims the conjectured bound.

Result: Proved(see note)
Status: Resolved
AI contribution: AI-discovered
Method: Argument
Field: Analysis of Boolean functions
Posed by: Michel Talagrand
Year posed: 1989
Years open: 37y
Solved: 2026-08-16
Model: Odin Automatic AI Research Agent
Vendor: —
Collaborators: Junwei Lu, Shengtao Guo, Ethan X. Fang
Verification: Unreviewed
Publication: Preprint
Significance: 37 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

The claim is the exact conjectured decay: $\psi_{\mu_a}(u) \le C_a/\sqrt{\log u}$ for every $u > 1$ and every $n$ , with $C_a$ dimension-free - concretely $\lesssim \kappa_a^2(\log\frac{\kappa_a}{\kappa_a-1})^{1/2}$ where $\kappa_a = (1+a)/(1-a)$ .

What is new is one step in a three-paper chain rather than a proof from scratch, and the paper is explicit about it. Chen's reverse-heat and Boolean-bridge framework and Xiang-Zhang's localized terminal-discrepancy method are taken as given; the addition is a power coupling that splits each reverse edge ratio into two geometric powers, producing a switched exponential weight that restores the exact reverse jump rate of the perturbed coordinate. Because the frozen exponent then has a fixed numerator, no growing stopping buffer is needed and the $\log\log u$ factor disappears. That last $\log\log$ is what stood between the previous work and Talagrand's statement.

What the AI did

The paper's disclosure is two sentences, in the abstract and again under a heading "The role of AI in this proof": "Odin Automatic AI Research Agent was used to discover the proof. The final proofs were reorganized by the authors."

Taken at face value, as this site's classification rule requires, that is an AI-discovered claim: the model produced the proof and the humans wrote it up. It is also thinner than any other entry at this tier. The paper says nothing about what Odin is, who builds it, which models it runs on, how it was steered, or how much of the manuscript is the agent's, and no public record of an "Odin Automatic AI Research Agent" could be found from this site - so the model maker field is left empty rather than guessed at.

The contribution being credited is a single idea: the power coupling that splits each reverse edge ratio into two geometric powers, which is what removes the iterated-logarithmic loss from the framework the paper inherits. The three named humans are Junwei Lu (Harvard T.H. Chan School of Public Health), Shengtao Guo and Ethan X. Fang.

Verification

A three-day-old arXiv preprint with no independent endorsement, so Unreviewed, and no mathematics was checked here.

The chain it sits in was checked, and it holds. Talagrand's hypercube conjecture was open - O'Donnell's problem list still carries it, and even the Gaussian special case was open as of 2012. Yuansi Chen (arXiv:2511.19374, Nov 2025) proved it up to a dimension-free $(\log\log)^{3/2}$ factor; Yanjin Xiang and Zhihua Zhang (arXiv:2606.04573, June 2026) cut that to $\log\log$ ; this paper claims to remove the loss entirely. Both predecessors exist, are by identifiable people, and say what this paper says they say, and the target bound matches Talagrand's own suggested $C_a/\sqrt{\log u}$ rather than something adjacent.

Two things cut the other way. Chen's paper needed a v2 to fix "a mistake in the previous draft which was kindly pointed out by Joseph Lehec" - that is what scrutiny in this corner looks like, and it is what this paper has not yet had. And the proof is attributed to an agent nobody outside the paper can identify, with no account of how it was run, so the process cannot be weighed either.

Sources

Submitted by VibeGene on 19 Aug 2026

Talagrand’s convolution conjecture

What was actually shown

What the AI did

Verification

Sources

Discussion