First-Extinction Law for Resampling Processes

We show that the first-extinction time in resampling processes is an observable governed by absorbing boundaries, and derive an exact and tractable expression for it. The computational cost of evaluating our formula scales linearly with the number of states $M$ and replaces the classical $2^M$ inclusion--exclusion terms, enabling the prediction of extinction times in systems with large state spaces and providing a unified quantitative law for extinction driven by sampling noise. This yields precise predictions for both genetic drift and the onset of model collapse.