Test supermartingales are complete for locally defined sequential hypotheses

E-processes give a general way to do anytime-valid sequential testing, while test supermartingales satisfy a simpler one-step conditional inequality under every law in the null. The two classes are not the same in general, so it is natural to ask when the extra freedom of an e-process helps for testing. We study sequential nulls described locally: after each finite history $s$, a set $\K(s)$ specifies the distributions allowed for the next observation. When the graph of $\K$ is analytic and the null contains all measurable sequential selections from these sets, we show that every composite e-process is pointwise dominated by a composite test supermartingale. Thus restricting to test supermartingales does not lose threshold-crossing power for this class of hypotheses. A fuller version with proofs and technical details is available in the full paper (link: https://dronakhurana.github.io/files/complete-class-ASH.pdf). The result does not require local domination, convexity, or compactness of the sets $\K(s)$. Examples range from bounded conditional means to nondominated, infinite-dimensional, and genuinely analytic conditional nulls.