= Expected Fourier weight after a random restriction
{title2=$\mathbb E_\rho W_{>s}(f_\rho)=\sum_S|\widehat f(S)|^2\mathbb P(\operatorname{Bin}(|S|,p)>s)$}
For a <random restriction>, first fix the live set $J$ and average over the unbiased assignment $a$ outside it. The coefficient at $U\subset J$ is $\sum_{B\subset J^c}\widehat f(U\cup B)\chi_B(a)$. Orthogonality eliminates cross terms in its squared expectation. Averaging over $J$ gives the displayed identity. The original <Fourier weight> is weighted by a <binomial distribution> survival probability, not simply preserved.
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