The Persistence–Feasibility Framework for Rare Events in Finite Systems

This work develops a unified framework for understanding the practical feasibility of rare stochastic events in finite physical systems. While classical probability guarantees eventual occurrence for events with non-zero probability, such results implicitly assume that the conditions enabling those events persist indefinitely. Real systems violate this assumption: enabling structures degrade, reset, or decohere before sufficient probability can accumulate.

The framework introduces three key quantities — the critical lifetime threshold, the coherence time of the enabling system, and a dimensionless persistence ratio — which together determine whether a system can support event accumulation. Feasibility is shown to arise from a competition between processes that build probability and processes that destroy it.

Extensions are developed for partially persistent systems, where structural memory is retained across resets, and for multi-channel systems with correlated destruction. These generalizations provide analytical conditions under which memory and parallelism can rescue feasibility.

The framework offers a consistent interpretive lens linking competing-hazard theory, renewal processes, and first-passage dynamics, and is applied across domains including prebiotic chemistry, quantum decoherence, cosmological fluctuations, and engineering reliability.