  Glasses, unlike their crystalline counterparts, exhibit low-frequency
nonphononic excitations whose frequencies $\omega$ follow a universal
$\mathcal{D}\!\left(\omega\right)\!\sim\!\omega^4$ density of states. The
process of glass formation generates positional disorder intertwined with
mechanical frustration, posing fundamental challenges in understanding the
origins of glassy nonphononic excitations. Here we suggest that \emph{minimal
complexes} -- mechanically-frustrated and positionally-disordered local
structures -- embody the minimal physical ingredients needed to generate
glasslike excitations. We investigate the individual effects of mechanical
frustration and positional disorder on the vibrational spectrum of isolated
minimal complexes, and demonstrate that ensembles of marginally stable minimal
complexes yield $\mathcal{D}\!\left(\omega\right)\!\sim\!\omega^4$.
Furthermore, glasslike excitation emerge by embedding a single minimal complex
within a perfect lattice. Consequently, minimal complexes offer a conceptual
framework to understand glasslike excitations from first principles, as well as
a practical computational method for introducing them into solids.
