Emergence of Finite-Time Singularities from Accelerated Event Recurrence: Insights into the Mechanism of Catastrophic Failure

We develop a discrete-event modeling framework that captures the progression of geophysical systems toward catastrophic failure through sequences of distinct damage events. By representing system evolution as a succession of temporally accelerating and amplitude-varying events, the framework reveals how finite-time singularities, both logarithmic and power law types, naturally emerge from the interplay between shrinking interevent intervals and growing event magnitudes. This event-based perspective provides an intuitive physical understanding of rupture processes, highlighting how precursory signals such as accelerating strain rate, event frequency, and energy release can be traced back to simple underlying mechanisms. A mean-field formulation further links the observed power law exponents to the evolving stiffness of the system under constant or time-varying stress. Incorporating stochastic fluctuations, the model captures the inherent randomness of natural systems leading to the emergence of stochastic finite-time singular behavior. Altogether, this unified approach offers a simple conceptual and quantitative tool for interpreting the lead-up to failure in a wide range of geophysical settings.