Bounded cumulative observables from local linear relaxation

Abstract

Cumulative observables often exhibit saturation in systems involving propagation or spreading with local dissipation. This work shows that bounded cumulative response follows directly from local linear relaxation. Linear cumulative observables accumulated over the lifetime of a relaxing signal are limited by a scale set by the relaxation time, independent of geometry, dimensionality, or microscopic transport dynamics. When relaxation is mapped to space through transport or spreading, this temporal bound yields a corresponding spatial saturation scale determined by the transport law. The result shows that cumulative saturation follows directly from exponential local relaxation and does not depend on the specific transport mechanism.