  Suppose we observe an infinite series of coin flips $X_1,X_2,\dots$, and wish
to sequentially test the null that these binary random variables are
exchangeable against Markovian alternatives. We utilize a geometric concept
called "fork-convexity" (an adapted analog of convexity) that lies at the heart
of this problem, and relate it to other concepts like Snell envelopes that are
absent in the sequential testing literature. By demonstrating that the
alternative lies within the \emph{fork-convex hull} of the null, we prove that
any nonnegative supermartingale under the exchangeable null is necessarily also
a supermartingale under the alternative, and thus yields a powerless test. We
then combine ideas from universal inference (maximum likelihood under the null)
and the method of mixtures (Jeffreys' prior over the alternative) to derive a
nonnegative process that is upper bounded by a martingale, but is not itself a
supermartingale. We show that this process yields safe e-values, which in turn
yield sequential level-$\alpha$ tests that are consistent (power one), using
regret bounds from universal coding to demonstrate their rate-optimal power. We
present ways to extend these results to any finite alphabet and to Markovian
alternatives of any order using a "double mixture" approach. We also discuss
their power against change point alternatives, and give general approaches
based on betting for unstructured or ill-specified alternatives.
