When do exact and powerful p-values and e-values exist?

Given a composite null $\mathcal P$ and composite alternative $\mathcal Q$, when and how can we construct a p-value whose distribution is exactly uniform under the null, and stochastically smaller than uniform under the alternative? Similarly, when and how can we construct an e-value whose expectation exactly equals one under the null, but its expected logarithm under the alternative is positive? We answer these basic questions, and other related ones, when $\mathcal P$ and $\mathcal Q$ are convex polytopes (in the space of probability measures). We prove that such constructions are possible if and only if (the convex hull of) $\mathcal Q$ does not intersect the span of $\mathcal P$. If the p-value is allowed to be stochastically larger than uniform under $P \in \mathcal P$, and the e-value can have expectation at most one under $P \in \mathcal P$, then it is achievable whenever $\mathcal P$ and $\mathcal Q$ are disjoint. The proofs utilize recently developed techniques in simultaneous optimal transport. A key role is played by coarsening the filtration: sometimes, no such p-value or e-value exists in the richest data filtration, but it does exist in some reduced filtration, and our work provides the first general characterization of when or why such a phenomenon occurs. We also provide an iterative construction that explicitly constructs such processes, that under certain conditions finds the one that grows fastest under a specific alternative $\mathcal Q$. We discuss implications for the construction of composite nonnegative (super)martingales, and end with some conjectures and open problems.