(4) It provides further theoretical insights for several previously proposed ad hoc choices made in the Spin-LSC algorithm, such as the procedure to randomly sample the mapping angle variables (Eq. 125). It also provides a rigorous initial condition sampling procedure through the SU (N ) mapping formalism, which is naturally connected to the distribution of the generalized Euler angle variables dΩ (Eq. 21), as well as new insights into the choice of the focused initial conditions that connects with the angle variables (Eq. 80 and Eq. 81).