The Gaussian propeller conjecture

For a measurable partition $A_1,\dots,A_k$ of $\mathbb R^d$ and the standard Gaussian measure $\gamma_d$ , let $z(A_i)=\int_{A_i}x\,d\gamma_d$ and $F=\sum_i\|z(A_i)\|^2$ . Khot and Naor, studying approximate kernel clustering and its Unique Games hardness, conjectured that for $k\ge3$ the maximum of $F$ is attained by the propeller: three planar sectors of angle $2\pi/3$ times an orthogonal factor, giving $9/(8\pi)$ . Heilman, Jagannath and Naor proved it in $\mathbb R^3$ . Is $\sum_i\|\int_{A_i}x\,d\gamma_d\|^2\le 9/(8\pi)$ for every finite partition in every dimension?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Gaussian geometry; partitions, kernel clustering
Posed by: S. Khot and A. Naor, Approximate kernel clustering, Mathematika 55 (2009)
Year posed: 2009
Years open: 17y
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: 30 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for all $d,k$ , every measurable partition of $\mathbb R^d$ satisfies $\sum_i\|\int_{A_i}x\,d\gamma_d\|^2\le9/(8\pi)$ , with equality for the propeller when $d\ge2$ , $k\ge3$ . The new step shows extremizers with five or more active cells cannot beat $9/(8\pi)$ , using Ehrhard's inequality and residual-score estimates; four-cell configurations come from Heilman-Jagannath-Naor. Consequence stated: combined with the release's Unique Games theorem, NP-hardness of beating the factor $(8\pi/9)(1-1/k)$ for identity-target kernel clustering, $k\ge3$ . Uniqueness of extremizers is not claimed.

What the AI did

The release README says all 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. 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.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the paper states it with the Khot-Naor reference. The proof uses the Heilman-Jagannath-Naor $\mathbb R^3$ theorem (computer-assisted) as an established input and was not refereed. The challenge ComparatorChallenges/GaussianPropeller.lean (OAI.GaussianPropeller.all_partitions, in lean/formalization.yaml) was read here: for all $d,k>0$ and every measurable a.e.-partition, $\sum_i\|\int_{A_i}x\,d\gamma\|^2\le9/(8\pi)$ , and for $d\ge2$ , $k\ge3$ the explicit propeller attains it. This states the headline. Not rebuilt here. The kernel-clustering hardness consequence is not formalized and depends on the release's separate Unique Games theorem.

The Gaussian propeller conjecture

What was actually shown

What the AI did

Verification

Sources

Discussion