Four distinct admissibility geometries govern sequential and distribution-free inference: Blackwell risk dominance over convex risk sets, anytime-valid admissibility within the nonnegative supermartingale cone, marginal coverage validity over exchangeable prediction sets, and Cesàro approachability (CAA) admissibility, which reaches the risk-set boundary via approachability-style arguments rather than explicit priors. We prove a criterion separation theorem: the four classes of admissible procedures are pairwise non-nested. Each geometry carries a different certificate of optimality: a supporting-hyperplane prior (Blackwell), a nonnegative supermartingale (anytime-valid), an exchangeability rank (coverage), or a Cesàro steering argument (CAA). Martingale coherence is necessary for Blackwell admissibility and necessary and sufficient for anytime-valid admissibility within e-processes, but is not sufficient for Blackwell admissibility and is not necessary for coverage validity or CAA-admissibility. All four criteria can be viewed through a common schematic template (minimize Bayesian risk subject to a feasibility constraint), but the decision spaces, partial orders, and performance metrics differ by criterion, making them geometrically incompatible. Admissibility is irreducibly criterion-relative.

翻译：四种不同的可容许性几何支配着序贯和无分布推断：凸风险集上的布莱克威尔风险占优、非负上鞅锥内的随时有效可容许性、可交换预测集上的边际覆盖有效性，以及切萨罗逼近可容许性——后者通过逼近式论证而非显式先验达到风险集边界。我们证明了一个判据分离定理：四类可容许程序两两不嵌套。每种几何携带不同的最优性证书：支撑超平面先验（布莱克威尔）、非负上鞅（随时有效）、可交换性秩（覆盖）或切萨罗导向论证（CAA）。鞅一致性对于布莱克威尔可容许性是必要的，对于e-过程中的随时有效可容许性既是必要的也是充分的，但对于布莱克威尔可容许性并非充分，对于覆盖有效性或CAA可容许性也非必要。所有四种判据均可通过一个共同的示意模板（在可行性约束下最小化贝叶斯风险）来审视，但不同判据的决策空间、偏序和性能度量各异，导致其在几何上不相容。可容许性不可化约地依赖于判据。