PDF is constructed for each star, and we use the mode of the distribution $DM_{\rm peak}$ and interval between the 84% and 16% percentiles, $\Delta DM = \left(DM_{84}-DM_{16}\right)/2$, as the 1$\sigma$ uncertainty. We find $N = 5814$ K giants cross-matched with the SDSS-*Gaia* proper motion sample. Thus, our resulting K giant sample has 3D positions (with distance moduli described using a Gaussian PDF), 3D velocities and spectroscopic metallicities.

Sagittarius Stream {#sec:sgr}
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![image](sgr_pms.png){width="16cm" height="13.71cm"}

![Proper motions in Galactic coordinates ($\mu_l$ left panels, $\mu_b$ right panel) against longitude in the Sagittarius (Sgr) coordinate system. The symbols and colors are identical to Fig. \[fig:sgr\_pms\]. Here, we have zoomed in on the regions of the Sgr stream that have proper motion constraints in the literature. The SDSS-*Gaia* RRL proper motions in the Sgr stream (solid black squares) are in excellent agreement with the literature values.[]{data-label="fig:pm_comp"}](pm_comp_sgr.png){width="8.5cm" height="8.5cm"}

Before introducing our model for halo rotation, we identify RRL stars in our sample that likely belong to the Sagittarius (Sgr) stream. This vast substructure is very prominent in the SDSS footprint [@belokurov06], and thus it may overwhelm any halo rotation signatures associated with earlier accretion events. Furthermore, previous works have independently measured proper motions of Sgr 