Author verified AI-led

Talagrand's convolution conjecture proved on the Boolean hypercube

Convolution by a measure on the Boolean hypercube obeys the dimension-free weak-type decay Talagrand conjectured in 1989, removing the iterated-logarithm factor carried by the best previous bound.

Lab: Independent
Model: Odin Automatic AI Research Agent
Field: Mathematics
Date: 2026-08-16
Human collaborators: Junwei Lu, Shengtao Guo, Ethan X. Fang
Problem posed: 1989 · open 37 yrs

Sources

Original work

arXivWeak-Type Bounds for Convolution on the Boolean Hypercube↗arXivChen, Talagrand's convolution conjecture up to loglog (prior bound)↗

Independent commentary

arXivShaposhnikov, Some remarks on Talagrand's convolution conjecture↗

What was found

For the Boolean hypercube with uniform measure and convolution by a finite positive measure, Talagrand conjectured that the anti-concentration profile decays like $1/\sqrt{\log u}$ , with a constant free of the dimension. Yuansi Chen reached that decay in November 2025 up to an additional $\log\log$ factor, using a perturbed reverse heat flow. This paper reworks the endpoint comparison so that the exponent has a fixed numerator, which removes the need for a growing stopping buffer and with it the $\log\log$ factor, giving the conjectured bound.

Novelty check

Talagrand posed the hypercube conjecture in 1989 and it remained on O'Donnell's open-problem list; even the Gaussian special case was open as of 2012. The prior state of the art is fixed precisely by Chen, arXiv:2511.19374, November 2025, which proves the bound up to a $\log\log$ correction and says so in its title. This paper's stated contribution is the removal of exactly that factor, and the mechanism it credits, a fixed-numerator frozen exponent that dispenses with the growing stopping buffer, is specific to the gap. The result is one step in a chain rather than a proof from scratch, and it is new.

Caveats and known objections

A preprint days old at entry, unrefereed and not formalized. Alexander Shaposhnikov has since given a self-contained martingale proof of the Boolean case with explicit dimension-free constants, which corroborates the theorem by a different method but is not a check of this paper's argument. The AI disclosure is two sentences, in the abstract and again under a heading 'The role of AI in this proof': the Odin Automatic AI Research Agent was used to discover the proof, and the final proofs were reorganized by the authors. Autonomy graded ai-led on that statement taken at face value, but the claim cannot be audited: Odin is not a publicly documented system, no vendor is named, no transcript or artifact is published, and there is no way to separate discovery from reorganization. Read the model field as the authors' attribution rather than as a verified fact.

Independent checks

Alexander Shaposhnikov (independent second proof): Gave self-contained martingale proofs of the convolution conjecture in the Gaussian and Boolean settings with explicit dimension-free constants, by a stopped density martingale rather than the reverse-heat approach, and credits Lu, Guo and Fang with settling the Boolean case. · link ↗

The open problem

MathDBTalagrand's convolution conjecture for the Boolean hypercube

371652

Nothing mechanically. It records what a problem says, what is known about it, and the standing of any claimed solution.

Also recorded at

vibemathedtalagrand-s-convolution-conjecture

Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.

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History of this entry

2026-09-24CheckedRecorded Alexander Shaposhnikov's independent martingale proof of the Boolean case. It confirms the theorem by another route rather than checking this proof, so no grade moved. source ↗
2026-08-21AddedEntered the registry graded Author verified and AI-led.

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Graded author verified for verification and ai-led for autonomy. What these mean.

Cite this entry

Plain text

whataifound.org. (2026). Talagrand's convolution conjecture proved on the Boolean hypercube. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-16-talagrand-convolution

BibTeX

@misc{whataifound-independent-2026-convolution,
  title        = {Talagrand's convolution conjecture proved on the Boolean hypercube},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by Independent. Verification: Author verified. Autonomy: AI-led.},
  url          = {https://whataifound.org/finding/2026-08-16-talagrand-convolution}
}

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