A Proof of the Third Borwein Conjecture

We study the coefficients of $S_n(q)=\prod_{j=1}^n\prod_{s=1}^4(1-q^{5j-s})$. An effective four-peak analysis gives the sign pattern predicted by the third Borwein conjecture for every $n\ge1750$ and every coefficient. The essential cancellation in residue classes $3$ and $4$ is retained as an exact