Abstract

Predictive systems may combine a predictor, a sequential monitor, a prediction set, and an online strategy, each governed by a different optimality criterion. We study when a guarantee for one component can be transferred to another. Four admissibility geometries shape sequential and distribution-free inference: Blackwell risk dominance over convex risk sets; anytime-valid admissibility within the $e$-process class; fixed-level marginal coverage with expected-length efficiency within a declared rank-indexed family under exchangeability; and choice-based approachability (CApp) boundary-feasibility, under which each parameter’s time-averaged risk reaches its oracle boundary value in Cesàro average. Each geometry carries a distinct certificate of optimality: a supporting-hyperplane prior, a nonnegative martingale in the point-null anytime-valid setting, a conformal exchangeability rank, and a Cesàro steering argument, respectively. We embed all four in a common ambient space, the space $\Sigma$ of predictive systems: tuples $\Delta=(\delta,E,\hat C,\sigma)$ consisting of a predictor, an $e$-process, a prediction set, and an online strategy. Within $\Sigma$, we prove a witness-based non-nesting theorem: for each ordered pair of the four criterion classes, an explicit predictive system, active in both relevant coordinates, lies in one class but not the other. The result establishes non-nesting across different object spaces and partial orders; it does not claim that the four criteria are practically incompatible. Bayesian posterior predictive means under a single prior are martingales under their prior predictive law. For a point null, anytime-valid admissibility within $e$-processes is equivalent to the nonnegative martingale property. By contrast, self-consistency under a predictor’s own law does not imply Blackwell admissibility, as demonstrated by a Bernoulli log-loss counterexample; neither coverage validity nor CApp boundary-feasibility requires a martingale structure. All four criteria fit a common design template: specify the criterion’s feasibility constraint, then optimize Bayesian integrated risk within it. Their decision spaces, partial orders, and risk functionals nevertheless remain distinct. Admissibility is criterion-relative.