An assignment move is $v_it_i$ or $v_if_i$ with $i<n$. A pass move is $v_it_i$ with $n\leq i<R$, or $y_iz_i$. A regular position is reachable from the initial graph by such moves. Write $r$ and $k$ for the numbers of surviving $v_i$ and $y_i$, respectively.

Claim 7: when $m$ is odd, $r\geq m$, and $k\geq1$, any other move loses unless it is $sa_j$ and all literal vertices of clause $j$ are gone. Claim 8: at a regular position with $r\geq m$, such an exceptional move wins exactly when $r+k$ is even.