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 share a common optimization template (minimize Bayesian risk subject to a feasibility constraint), but the constraint sets operate over different spaces, partial orders, and performance metrics, 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列化与无分布推断：凸风险集上的布莱克韦尔风险优势、非负上鞅锥内的任意时间有效可容许性、可交换预测集上的边际覆盖有效性，以及通过逼近论式论证（而非显式先验）达到风险集边界的切萨罗逼近可容许性。我们证明了一个准则分离定理：这四类可容许程序两两互不包含。每种几何对应不同的最优性证明：支撑超平面先验（布莱克韦尔）、非负上鞅（任意时间有效）、可交换秩（覆盖）或切萨罗引导论证。鞅相干性是布莱克韦尔可容许性的必要条件，也是e-过程内任意时间有效可容许性的充要条件，但不足以保证布莱克韦尔可容许性，亦非覆盖有效性或切萨罗逼近可容许性的必要条件。所有四种准则共享一个共同的优化模板（在可行性约束下最小化贝叶斯风险），但约束集作用于不同的空间、偏序和性能度量，导致它们在几何上互不相容。可容许性本质上是准则相对的。