The Gaussian propeller conjecture

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.

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.