> How is p uniformly distributed under H0? That's the very definition of a p-value! The mapping of data to p-values is chosen to have a uniform distribution of

> How is p uniformly distributed under H0? That's the very definition of a p-value! The mapping of data to p-values is chosen to have a uniform distribution of p-values when the data is distributed according to the null hypothesis. That's the property that makes p-values interesting.