Small Type I sums for binary digit parity ($N^{-1}{\sum }_{m\le N^{1/100}}|{\sum }_{n\in I_m}(-1{)}^{s_2(mn)}|\ll N^{\sigma +1/100-1}$)

The additive character orthogonality filter $1_{m\mid r}=m^{-1}{\sum }_{a=0}^{m-1}e(ar/m)$ restricts the uniform interval cancellation for binary digit parity to multiples of $m$, without increasing its bound. Each inner Type I sum is therefore $O(N^{\sigma })$. Summing the at most $N^{1/100}$ outer terms proves the displayed estimate, which is smaller than every fixed negative power of $\log N$ for large $N$.