Low-temperature properties of crystalline solids can be understood using harmonic perturbations around a perfect lattice, as in Debye's theory. Low-temperature properties of amorphous solids, however, strongly depart from such descriptions, displaying enhanced transport, activated slow dynamics across energy barriers, excess vibrational modes with respect to Debye's theory (i.e., a boson peak), and complex irreversible responses to small mechanical deformations. These experimental observations indirectly suggest that the dynamics of amorphous solids becomes anomalous at low temperatures. Here, we present direct numerical evidence that vibrations change nature at a well-defined location deep inside the glass phase of a simple glass former. We provide a real-space description of this transition and of the rapidly growing time-and lengthscales that accompany it. Our results provide the seed for a universal understanding of low-temperature glass anomalies within the theoretical framework of the recently discovered Gardner phase transition.glass transition | disordered solids | Gardner transition | computer simulations | hard spheres U nderstanding the nature of the glass transition, which describes the gradual transformation of a viscous liquid into an amorphous solid, remains an open challenge in condensed matter physics (1, 2). As a result, the glass phase itself is not well understood either. The main challenge is to connect the localized, or "caged," dynamics that characterizes the glass transition to the low-temperature anomalies that distinguish amorphous solids from their crystalline counterparts (3-7). Recent theoretical advances, building on the random first-order transition approach (8), have led to an exact mathematical description of both the glass transition and the amorphous phases of hard spheres in the mean-field limit of infinite-dimensional space (9). A surprising outcome has been the discovery of a novel phase transition inside the amorphous phase, separating the localized states produced at the glass transition from their inherent structures. This Gardner transition (10), which marks the emergence of a fractal hierarchy of marginally stable glass states, can be viewed as a glass transition deep within a glass, at which vibrational motion dramatically slows down and becomes spatially correlated (11). Although these theoretical findings promise to explain and unify the emergence of low-temperature anomalies in amorphous solids, the gap remains wide between mean-field calculations (9, 11) and experimental work. Here, we provide direct numerical evidence that vibrational motion in a simple 3D glass-former becomes anomalous at a well-defined location inside the glass phase. In particular, we report the rapid growth of a relaxation time related to cooperative vibrations, a nontrivial change in the probability distribution function of a global order parameter, and the rapid growth of a correlation length. We also relate these findings to observed anomalies in low-temperature laboratory glasses. These re...