Appendix 1—figure 2. Family of kernel functions $f_k(\omega )$ and $g_k(\omega )$ given in Equation 26a.

These functions represent weights of the coarse-graining kernels Equation 22a and Equation 22b that are defined such that the kernels satisfy the consistency relation Equation 19. $\omega =\text{acos}(\mathbf{r}\cdot {\mathbf{r}}^{\prime })$ denotes angular distances between $\mathbf{r}$ and.${\mathbf{r}}^{\prime }$ Coarse-graining of a conserved number of particles on a sphere to determine a density field $\rho$ (Equation 2a, main text) requires a different weighting – $f_k(\omega )$ – than the coarse-graining of an associated flux $\mathbf{J}$ (Equation 2b , main text), which requires a weighting $g_k(\omega )$ instead to ensure that coarse-grained fields obey mass conservation Equation 18. A characteristic coarse-graining length scale associated with these kernels is the half-width at half-maximum (HWHM), which is related to $k$ by HWHM $=\arccos (2^{-1/k})$.