Understanding the limits of growth in finite systems remains a central challenge across physics, energy systems, and ecological economics. In this work, we develop a general mathematical framework for systems governed by iterative processes with multiplicative costs, showing that the fraction of useful resources decays exponentially with the number of cycles. In closed systems, we prove that the accumulation of positive costs leads inevitably to the asymptotic exhaustion of useful resources, independently of initial resource magnitude. As a consequence, the operational lifetime scales only logarithmically with the available resource. We incorporate efficiency improvements and demand growth, showing that when demand increases faster than efficiency, systems enter a regime of accelerated depletion, providing a formal formulation of a multiplicative versionof the Jevons paradox. We then extend the framework to open systems with external resource access. Even in the presence of abundant external resources, growth remains fundamentally constrained by finite access rates imposed by physical limits such as transport and extraction. In this setting, sustainability requires convergence toward a dynamic balance between consumption and supply.These results support a unified interpretation: limits to growth arise not only from resource finiteness, but from the cumulative structure of costs and the bounded nature of resource accessibility. In the asymptotic regime, systems tend toward a state in which available energy is entirely devoted to sustaining their own operation, leading to the effective disappearance of useful work. This framework provides a general theoretical basis for understanding growth limits across physical, energetic, and technological systems, independent of specific model assumptions.
Enrique Moreno (Sun,) studied this question.