Deletion Optimality Characterizes Shared Gumbel Sampling

Shared randomness coordinates categorical samples across different distributions, supporting correlated sampling and counterfactual generation. What makes shared Gumbel sampling a distinguished choice among exact sampling rules? We show that a simple deletion principle determines its entire joint behavior. On a finite alphabet with at least three labels, any exact shared-randomness sampler that max- imizes agreement between every distribution and its one-label deletions has the same finite-dimensional output laws as shared Gumbel sampling. Thus, optimiz- ing agreement on these local pairs uniquely determines the coupling across all distributions. As a consequence, no exact shared-randomness sampler can match or improve Gumbel agreement on every distribution pair and strictly improve it on any pair. The characterization covers arbitrary measurable samplers. Its proof derives a common race representation from deletion optimality and yields quanti- tative certificates relating comparison reversals to deletion deficits. On countably infinite alphabets, we construct distinct deletion-optimal couplings within a com- mon tail class and show that continuity under prefix truncation restores unique- ness. These results establish a local characterization of shared Gumbel sampling and identify the boundary between finite rigidity and countable freedom.