I second Wei's question. I can imagine doing logical proofs about how your successor's algorithms operate to try to maximize a utility function relative to a lawfully updated epistemic state, and would consider my current struggle to be how to expand this to a notion of a lawfully approximately updated epistemic state. If you say 'martingale' I have no idea where to enter the problem at all, or where the base statistical guarantees that form part of the martingale would come from. It can't be statistical testing unless the problem is i.i.d. because otherwise every context shift breaks the guarantee.