Identification of flexible local environments from a disordered medium has been a long-standing challenge. Here, we introduce a time-relevant structural Shannon entropy as a unique feature of the atomic-scale environment in glass, which is based on a metric of the time-invariant, or ergodic, and Voronoi structural diversity that an atom experiences during a sufficiently long-time thermal fluctuation. This new concept of time-relevant Shannon entropy simultaneously integrates the static topology and the vibrational feature such that it potentially probes all the possible configurational space in a sub-basin of the local potential energy landscape. This structural representation is not only capable of predicting the energy barrier of an elementary structural excitation but also demonstrates a robust correlation with the boson peak in metallic glasses, although the physical entity is defined from a purely structural aspect. The proposition, therefore, represents a successful demonstration of the physics-informed structure-property relationship in amorphous materials.