 Sign rounding bound for a unit eigenvector

Suppose $v_i=\pm 1/\sqrt{n}$ encodes the true partition and $\hat{v}$ is a unit estimated eigenvector. Every coordinate of the wrong sign contributes at least $1/n$ to ${\min }_{\sigma =\pm 1}\Vert \hat{v}-\sigma v{\Vert }_2^2=2(1-|{\hat{v}}^{\top }v|)$. Since this is at most $2(1-|{\hat{v}}^{\top }v{|}^2)$, which is the squared Frobenius distance between the rank-one orthogonal projection matrices, the bound follows. Coordinates estimated as zero may be assigned consistently to either group.