  We introduce a weak order ideal property that suffices for establishing the
Evans-Griffith Syzygy Theorem. We study this weak order ideal property in
settings that allow for comparison between homological algebra over a local
ring $R$ versus a hypersurface ring $R/(x^n)$. Consequently we solve some
relevant cases of the Evans-Griffith syzygy conjecture over local rings of
unramified mixed characteristic $p$, with the case of syzygies of prime ideals
of Cohen-Macaulay local rings of unramified mixed characteristic being noted.
We reduce the remaining considerations to modules annihilated by $p^s$, $s>0$,
that have finite projective dimension over a hypersurface ring.
