Universal entanglement embezzlement and divergent nonlocal magic from generic local chaotic quantum evolution

We show that, starting from a product state, local unitary quantum evolutions generate intermediate states which exhibit a multiscale entanglement-spectrum structure required for universal entanglement embezzlement. This facilitates entanglement extraction from a catalyst many-body state while leaving it asymptotically unchanged. Remarkably, these atypical structures emerge generically at intermediate stages, well before reaching maximum entropy where thermalization has flattened out the spectral hierarchy. The resulting state is accompanied by nonlocal nonstabilizerness that diverges with the system size, consistent with a recently established equivalence between universal embezzlement and divergent nonlocal magic. Thus, without any fine tuning, a chaotic quantum evolution generates intermediate states which form a universal family of catalytic reservoirs.