Steinhaus first-moment lower bound (${\sum }_i|a_i{|}^2\le 2(\mathbb{E}|{\sum }_i{a}_i{\eta }_i|{)}^2$)

For independent Steinhaus random variables, condition on their acute angles and use the sharp Rademacher second-moment inequality on the $2d$ sign coefficients $a_i\cos {\theta }_i,i{a}_i\sin {\theta }_i$. Their squared moduli sum to ${\sum }_i|a_i{|}^2$ independently of the angles, so the conditional first moment is at least the same fixed square-root bound. Averaging proves the assertion.