  Radioactive decay of potassium (K), thorium (Th), and uranium (U) power the
Earth's engine, with variations in 232Th/238U recording planetary
differentiation, atmospheric oxidation, and biologically mediated processes. We
report several thousand $^{232}$Th/$^{238}$U ($\kappa$) and time-integrated Pb
isotopic ($\kappa$$_{Pb}$) values and assess their ratios for the Earth, core,
and silicate Earth. Complementary bulk silicate Earth domains (i.e.,
continental crust $\kappa_{Pb}^{CC}$ = 3.94 $^{+0.20}_{-0.11}$ and modern
mantle $\kappa_{Pb}^{MM}$ = 3.87 $^{+0.15}_{-0.07}$, respectively) tightly
bracket the solar system initial $\kappa_{Pb}^{SS}$ = 3.890 $\pm$ 0.015. These
findings reveal the bulk silicate Earth's $\kappa$$_{Pb}^{BSE}$ is 3.90
$^{+0.13}_{-0.07}$ (or Th/U = 3.77 for the mass ratio), which resolves a
long-standing debate regarding the Earth's Th/U value. We performed a Monte
Carlo simulation to calculate the $\kappa_{Pb}$ of the BSE and bulk Earth for a
range of U concentrations in the core (from 0 to 10 ng/g). Comparison of our
results with $\kappa$$_{Pb}^{SS}$ constrains the available U and Th budget in
the core. Negligible Th/U fractionation accompanied accretion, core formation,
and crust - mantle differentiation, and trivial amounts of these elements (0.07
ppb by weight, equivalent to 0.014 TW of radiogenic power) were added to the
core and do not power the geodynamo.
