Every lawful wait, translated onto one ruler.

Because $G$ is discrete, its randomised probability-integral transform is

$U=F(G-1)+V\,P(G),\qquad V\sim\operatorname{Uniform}(0,1).$

Under the null, $U$ is uniform. The laboratory checks the pooled uniform distribution and lag-one residual correlation. It does not pretend residuals from different balls are independent: every null run simulates complete $k$-of-$N$ draws, recomputes both statistics, and corrects their minimum p-value.

The final open wait is censored. For every gap that does finish, the transform therefore uses the geometric law conditional on completion before the recorded history ends. Long unfinished waits are not silently discarded from an untruncated reference law.