  This article establishes cutoff stability also known as abrupt thermalization
for generic multidimensional Hurwitz stable Ornstein-Uhlenbeck systems with
(possibly degenerate) L\'evy noise at fixed noise intensity. The results are
based on several ergodicity quantitative lower and upper bounds some of which
make use of the recently established shift linearity property of the
Wasserstein-Kantorovich-Rubinstein distance by the authors. It covers such
irregular systems like Jacobi chains and more general networks of coupled
harmonic oscillators with a heat bath (including L\'evy excitations) at
constant temperature on the outer edges and the so-called Brownian gyrator.
