 Adjacent count tails in an exact Poisson confidence interval

For a count $s>0$ from a Poisson distribution, equal-tail inversion uses the inclusive upper-tail event $S\ge s$ for the lower endpoint and the inclusive lower-tail event $S\le s$ for the upper endpoint. Their cumulative distribution indices differ by one, as displayed. If $s=0$, the lower endpoint is zero. For $n$ observations of mean $\mu$, apply this to their total of mean $\lambda =n\mu$ and divide endpoints by $n$. The construction has at least the nominal frequentist coverage; discreteness can make it conservative.