Finite spin-bath coherence recurrence ($z_N(t_j)\to 1\text{for some }{t}_j\to \infty$)

For any finite set of couplings in $z_N(t)={\prod }_k\cos (2g_{k}t)$, the simultaneous Dirichlet approximation theorem supplies arbitrarily late near-alignment of all phases, or an exact common period. Thus finite bath coherence does not tend to zero as time tends to infinity. This remains true for almost every realization of continuous random couplings; an ensemble mean can decay while each realization has recurrences.