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Gigli's conjecture: Alexandrov curvature bounds are characterized by his distributional sectional curvature on noncollapsed RCD spaces

On an RCD(K,N)\mathrm{RCD}(K,N) metric measure space, Gigli (2019) defined a distributional Riemann curvature tensor R(X,Y,Z,W)R(X,Y,Z,W) acting on test vector fields and test functions. Let n≥2n\ge2, κ∈R\kappa\in\mathbb R and (M,d)(M,d) complete and separable. Gigli conjectured that (M,d)(M,d) is an nn-dimensional Alexandrov space of curvature at least κ\kappa if and only if (M,d,Hn)(M,d,\mathcal H^n) has full support, is RCD((n−1)κ,n)\mathrm{RCD}((n-1)\kappa,n), and satisfies R(X,Y,Y,X)(f)≥κ∫f(∣X∣2∣Y∣2−⟨X,Y⟩2) dHnR(X,Y,Y,X)(f)\ge\kappa\int f(|X|^2|Y|^2-\langle X,Y\rangle^2)\,d\mathcal H^n for all test fields X,YX,Y and nonnegative test functions ff. Is this characterization true?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Metric geometry; RCD spaces and Alexandrov spaces
Posed by
Nicola Gigli, Conjecture 1.1 of 'Riemann curvature tensor on RCD spaces and possible applications' (2019)
Year posed
2019
Years open
7y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
18 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every integer n≥2n\ge2 and κ∈R\kappa\in\mathbb R, a complete separable metric space is an nn-dimensional Alexandrov space with curvature at least κ\kappa if and only if, with m=Hnm=\mathcal H^n, it is a full-support unreduced RCD((n−1)κ,n)\mathrm{RCD}((n-1)\kappa,n) space whose distributional sectional curvature is at least κ\kappa in Gigli's original global test classes. The reference measure must be exactly Hausdorff measure; collapsed or weighted spaces are not covered, and dimension one is excluded. The companion proves that weak Hessian upper bounds hold along every minimizing geodesic in any full-support RCD(K,N)\mathrm{RCD}(K,N) space, 1<N<∞1<N<\infty.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. 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, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript uses a companion paper of the same date (weak Hessian bounds along every geodesic in RCD spaces) as an input to the converse direction.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Gigli's Conjecture 1.1. The only formal statement in the family is ComparatorChallenges/WeakHessian.json (OAI.WeakHessian.every_geodesic, listed in formalization.yaml), which formalizes the companion's lemma: a weak Hessian upper bound on a full-support RCD(K,N) space passes to a distributional bound along every constant-speed minimizing geodesic. That is one input to the converse direction, not the characterization, so under the tier rule this entry is unreviewed. The rest of the argument (forward direction, heat-regularized index inequality, frames along paths) is unformalized.

Sources

Changelog1 change

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