= Persistent-torque criterion for asymptotic spin alignment
{title2=$\int_0^\infty K_2(t)\,dt=\infty$}
For nonzero $J_h,J_t$ and a noncollinear initial configuration, the alignment equation integrates to $\operatorname{artanh}c(t)=\operatorname{artanh}c(0)+J_hJ_t\int_0^tK_2(s)\,ds$. 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 $\tanh1<1$. Exactly antiparallel initial spin and total angular momentum form a stationary, nongeneric exception.
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