Mean-lifetime survival energy estimate ($p=\exp [-H/(\gamma v{\tau }_0)]$)

An unstable particle of rest lifetime ${\tau }_0$ traversing distance $H$ at constant speed $v$ has survival probability $p=\exp [-H/(\gamma v{\tau }_0)]$. Requiring a flight time no greater than one dilated mean lifetime selects $p\ge e^{-1}$. In the ultrarelativistic approximation $v\simeq c$, this gives $\gamma \gtrsim H/(c{\tau }_0)$ and $E\gtrsim m{c}^{2}H/(c{\tau }_0)$. It is a specified survival-probability criterion, not a sharp cutoff for individual random decays. Continuous energy loss requires integrating the decay probability along the path.