= Steinhaus first-moment lower bound
{c}
{title2=$\sum_i|a_i|^2\leq2(\mathbb E|\sum_i a_i\eta_i|)^2$}
For independent <Steinhaus random variables>, condition on their acute angles and use the <sharp Rademacher second-moment inequality> on the $2d$ sign coefficients $a_i\cos\theta_i,ia_i\sin\theta_i$. 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.
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