  We show that the measured intrinsic octupole moments of $^{220}$Rn,
$^{224}$Ra, and $^{226}$Ra constrain the intrinsic Schiff moments of
$^{225}$Ra$^{221}$Rn, $^{223}$Rn, $^{223}$Fr, $^{225}$Ra, and $^{229}$Pa. The
result is a dramatically reduced uncertainty in intrinsic Schiff moments.
Direct measurements of octupole moments in odd nuclei will reduce the
uncertainty even more. The only significant source of nuclear-physics error in
the laboratory Schiff moments will then be the intrinsic matrix elements of the
time-reversal non-invariant interaction produced by CP-violating fundamental
physics. Those matrix elements are also correlated with octupole moments, but
with a larger systematic uncertainty.
