6533b835fe1ef96bd129fd25

RESEARCH PRODUCT

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We show that the measured intrinsic octupole moments of $^{220}\mathrm{Rn}$, $^{224}\mathrm{Ra}$, and $^{226}\mathrm{Ra}$ constrain the intrinsic Schiff moments of $^{225}\mathrm{Ra}$, $^{221}\mathrm{Rn}$, $^{223}\mathrm{Rn}$, $^{223}\mathrm{Fr}$, $^{225}\mathrm{Ra}$, and $^{229}\mathrm{Pa}$. 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 noninvariant interaction produced by $CP$-violating fundamental physics. Those matrix elements are also correlated with octupole moments, but with a larger systematic uncertainty.