Normalize the first Fourier term by averaging over balanced predicates. If its supports have size at least $\ell$ and its coefficients have total squared mass at most one, its second moment is at most $q^\ell+2(N+1)e^{-Nq^2/2}$ for every $q\geq0$, where $N$ is the number of inputs to a predicate. Its upper tail at $a>0$ is bounded by this quantity divided by $a^2$.

This is the normalized version of the estimates in Lemma 4.5 and Corollary 4.6, with the concentration bound obtained by conditioning on balance. The choice $q=N^{-1/4}$ yields the required high-degree decay.