Critical Lifetime Threshold Theorem: A Formal Framework for the Minimum System Duration Required for Near-Zero Probability Events to Occur

I introduce the Critical Lifetime Threshold Theorem (CLTT), a formal framework characterising the minimum system duration — termed the critical lifetime n* — required for an event of near-zero probability p to achieve a specified likelihood of occurrence. We derive the threshold n* = ln(2) / p as the half-life of improbability: the minimum lifetime at which occurrence becomes more probable than non-occurrence. A strict dichotomy is proven: systems with lifetime n_life >> n* experience the event with near-certainty, while systems with n_life << n* almost surely do not. Applications to abiogenesis, cosmological physics (Boltzmann brains), and engineering reliability are demonstrated. The framework provides a unified quantitative boundary between feasible and infeasible occurrence of improbable event