OpenAI · 096-01 · 2026-09-24 · Convex and metric geometry
The Gaussian propeller bound in every dimension
preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026
Family 096. The Gaussian propeller conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most . In dimension at least two, three planar sectors of angle , extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor for identity-target kernel clustering with fixed on rational centered positive semidefinite inputs.
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 . For dimension at least two and at least three cells, three planar sectors of angle , extended by an orthogonal Euclidean factor, attain the bound.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 096: The Gaussian propeller conjecture in every dimension. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.