Remark: Family 096. The Gaussian propeller conjecture

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

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.