Admissible E-Value Merging Under Uncertain Input Validity

Suppose that a nonnegative random vector contains valid e-variables on an unknown member of a prescribed family of coordinate sets. The remaining coordinates are unrestricted, and arbitrary dependence is allowed. We characterize admissible merging rules under pointwise domination among all finite-valued nonnegative Borel rules. Every valid rule is dominated by a minimum of normalized affine functions, with one certificate supported on each possible valid set. For the convex hull of these certificates, validity is equivalent to intersection with specified coordinate faces, whereas domination reverses set inclusion. Admissibility holds exactly when every vertex is the unique point of intersection with at least one required face. This condition has an equivalent formulation in terms of inclusion-minimal coverage by the vertex supports. It yields a finite deletion procedure producing an admissible envelope that dominates the starting valid envelope, with strict improvement whenever a vertex is deleted. Every admissible rule is continuous, concave, and coordinatewise nondecreasing. The number of vertices is bounded by the number of inclusion-minimal validity sets, and this bound is sharp. Algebraic examples include asymmetric mergers and lower trimmed means. We also establish the existence of an admissible maximizer of expected logarithmic evidence under bounded alternatives and identify a sufficient condition for zero optimal growth.