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
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 one over the square root of 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, not formalized, and with no independent endorsement recorded. 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.
The open problem
MathDBTalagrand's convolution conjecture for the Boolean hypercube
371652
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Also recorded at
vibemathedtalagrand-s-convolution-conjecture
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Entry history (1 event)
- AddedEntered the registry graded Author verified and AI-led.
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Cite this entry
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-convolutionBibTeX
@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}
}