The Gaussian propeller conjecture in every dimension

What OpenAI claims

Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most 9/(8π). In dimension at least two, three planar sectors of angle 2π/3, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor (8π/9)(1-1/k) for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs. (Lean)

Paper

1. The Gaussian propeller bound in every dimension

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove the Gaussian propeller conjecture: for every finite measurable partition of a Euclidean space, the sum of the squared lengths of its Gaussian first moments is at most 9/(8π). For dimension at least two and at least three cells, three planar sectors of angle 2π/3, extended by an orthogonal Euclidean factor, attain the bound.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.