Innovation and obsolescence describe dynamics of ever-churning and adapting social and biological systems. We formalize the connection with a reduced model of the dynamics of the “space of the possible” (e.g. technologies, mutations, theories) to which agents (e.g. firms, organisms, scientists) couple as they grow, die, and replicate. We predict three regimes: the space is finite, ever growing, or a Schumpeterian dystopia in which obsolescence drives the system to collapse. We reveal a critical boundary at which the space of the possible fluctuates dramatically in size, displaying recurrent periods of minimal and of veritable diversity. When the space is finite, corresponding to physically realizable systems, we find surprising structure. As one prediction, we consider the density of agents near and away from the innovative frontier that we compare with distributions of firm productivity, covid diversity, and citation rates for scientific publications. Remarkably, our minimal model derived from first principles aligns with empirical examples. In more recent work, we extend the model to consider more general structures to the space of the possible and big data on the information footprint of firms. The extensions put to empirical test our framework for unifying innovation and obsolescence across fields.