Correlation energy of the spin-polarized uniform electron gas at high density

The correlation energy per electron in the high-density uniform electron gas can be written as ${E}c({r}ₛ,{ζ})={{λ}}₀({ζ})ln{r}ₛ+{{ε}}₀({ζ})+{{λ}}₁({ζ}){r}ₛln{r}ₛ+O({r}ₛ)$, where ${r}ₛ$ is the Seitz radius and ${ζ}$ is the relative spin polarization. We derive an expression for ${{λ}}₁({ζ})$ that is exact for any ${ζ}$, including the paramagnetic and ferromagnetic limits, ${{λ}}₁(0)$ and ${{λ}}₁(1)$, and discover that the previously published ${{λ}}₁(1)$ value is incorrect. We trace this error to an integration and limit that do not commute. The spin resolution of ${{λ}}₁({ζ})$ into contributions of electron pairs is also derived.