(3). Spin-LSC is similar to the linearized method derived in this work (Eq. 117 and Eq. 119) and we provide here a proof from first principle. The EOMs are justified through the quantum-classical Liouville equation, resulting in the EOMs in Eq. 117 and Eq. 119. The difference between spin-LSC and the current formalism are the initial conditions. In our current formalism, the sampling of the mapping variables is determined by the expression of dΩ in Eq. 21, whereas in spin-LSC, the mapping variables {q n , p n } are randomly sampled from a Gaussian distribution, then universally scaled such that Eq. 57 (total population constraint) is satisfied. For the focused initial conditions, both the current formalism and spin-LSC use Eq. 82. Numerically, we found that our current formalism and spin-LSC produce similar initial conditions, for both the sampled and the focused procedures.