Persistent-torque criterion for asymptotic spin alignment (${\int }_0^{\infty }{K}_2(t)dt=\infty$)

For nonzero $J_h,J_t$ and a noncollinear initial configuration, the alignment equation integrates to $\mathrm{artanh}c(t)=\mathrm{artanh}c(0)+J_h{J}_t{\int }_0^t{K}_2(s)ds$. Hence the positive dissipative torque aligns the hole with the total vector if its integrated strength diverges. A fixed positive coefficient is sufficient. Pointwise positivity alone need not suffice for time-dependent coefficients: with $J_h=J_t=1$, $K_1=0$, $K_2=e^{-t}$ and $c(0)=0$, the limiting cosine is $\tanh 1<1$. Exactly antiparallel initial spin and total angular momentum form a stationary, nongeneric exception.