We prove that the correlation of a depth-$d$
unbounded fanin circuit of size $S$ with parity
of $n$ variables is at most $2^{-\Omega(n/(\log S)^{d-1})}$.

Substantial revision of main proof.

We prove that the correlation of a depth-$d$
unbounded fanin circuit of size $S$ with parity
of $n$ variables is at most $2^{-\Omega(n/(\log S)^{d-1})}$.