The David-Semmes problem in higher codimension: bounded Riesz transforms force uniform rectifiability for 1 < n < d-1

Let $\mu$ be an $n$ -Ahlfors-David regular measure on $\mathbb R^d$ and $R_\mu$ the $n$ -dimensional Riesz transform with kernel $x/|x|^{n+1}$ . David and Semmes asked whether boundedness of $R_\mu$ on $L^2(\mu)$ implies that $\mu$ is uniformly $n$ -rectifiable. The answer is yes for $n=1$ (Mattila-Melnikov-Verdera in the plane, curvature methods in general) and for $n=d-1$ (Nazarov-Tolsa-Volberg, 2014), whose maximum-principle argument has no known higher-codimension analogue; Mas and Tolsa characterized uniform rectifiability through $\rho$ -variation, a stronger hypothesis. Is every $n$ -AD-regular measure on $\mathbb R^d$ with $L^2$ -bounded Riesz transform uniformly $n$ -rectifiable when $1<n<d-1$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Harmonic analysis; geometric measure theory, uniform rectifiability
Posed by: Guy David and Stephen Semmes
Year posed: 1991
Years open: 35y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 45 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.3: for integers $d\ge4$ and $2\le n\le d-2$ , an $n$ -AD-regular Radon measure on $\mathbb R^d$ whose hard-truncated Riesz transforms are uniformly bounded $L^2(\mu)\to L^2(\mu;\mathbb R^d)$ is uniformly $n$ -rectifiable, with constants depending only on $d,n,C_{AD},C_R$ ; the conclusion is given as uniform big pieces of Lipschitz images of balls. With the known cases $n=1$ and $n=d-1$ this answers the David-Semmes question in every dimension and codimension. The converse direction and non-integer dimensions are not addressed here.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The single manuscript (September 24, 2026) is the whole family.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.3 was read against the David-Semmes question in the range left open; for $d\ge4$ and $2\le n\le d-2$ (all of $1<n<d-1$ ) it gives uniform rectifiability from a uniform $L^2$ bound on hard truncations, with no principal-value or variation hypothesis. The proof was not refereed. lean/formalization.yaml lists RieszQuantitative (OAI.RieszRectifiability.quantitative_higher_codimension_riesz_rectifiability). The comparator statement was read here: for such $d,n$ and constants $C_{AD}$ , $C_R$ there are $\theta>0$ and $M$ such that every regular $n$ -AD-regular measure with truncations bounded by $C_R$ has, at every support point and admissible radius, an $M$ -Lipschitz image of the $n$ -ball of radius $r$ carrying mass at least $\theta r^n$ in the ball. This states the headline (big pieces of Lipschitz images, the manuscript's definition of uniform rectifiability). Not rebuilt here.

The David-Semmes problem in higher codimension: bounded Riesz transforms force uniform rectifiability for 1 < n < d-1

What was actually shown

What the AI did

Verification

Sources

Discussion