The Gaussian propeller bound

OpenAI's proof (2026) of the Gaussian propeller conjecture of Khot and Naor: the Gaussian first moments of the cells of any finite measurable partition of a Euclidean space have squared lengths summing to at most 9/(8 pi), with equality for three planar sectors of angle 2 pi/3. Duality with Gaussian maxima, half-space and spherical-cap rearrangement, Ehrhard concavity of gap probabilities, residual covariances and a probability budget with exact margins; a sharp inequality for Gaussian maxima and the threshold for kernel clustering.

Six lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Gaussian first moments of partitions
2. Rearrangement bounds for one cell
3. Ehrhard concavity and the loss of one score
4. Two competing scores
5. Five or more cells and the propeller bound
6. Kernel clustering and the Gaussian constant

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Gaussian heat semigroups and a gradient bound (Ehrhard's inequality for convex sets)
• Gaussian vectors, covariance and independence.
• For the last lesson: Unique Games instances and polynomial-time reductions.