Exclusive-or gadget in Valiant's permanent reduction

The Exclusive-or gadget in Valiant's permanent reduction cancels invalid cycle covers of a directed graph using signed edge weights. In the original four-vertex junction matrix $X$, only vertices one and four have external connections. Writing $X(R;C)$ for row and column deletions, its permanents satisfyandSurviving weighted routes therefore enter once and leave at the opposite port. Invalid patterns have total weight zero, even when individual covers exist.^[ref]