The Gaussian and Arithmetic Price of Exact Pivotality for E-Values

This investigates the sharp asymptotic cost of imposing exact pivotality on e-values, measured through the loss of optimal expected logarithmic evidence under a simple alternative. For independent and identically distributed observations with finitely many competing null distributions, the paper establishes an exact additive information-price theorem in the nonarithmetic regime. Under the stated moment and covariance conditions, the optimality gap converges to I(B; W), the mutual information between the index of the minimum and the minimum value of a limiting Gaussian vector. The result includes a converse for arbitrary randomised output kernels and constructions achieving exact common null distributions and unit null expectations at every sample size. The paper further develops finite-alphabet arithmetic classifications, identifies persistent lattice-induced obstructions, establishes sharp results for two and three competing null distributions, and derives constructive bounds for higher-dimensional settings. Additional contributions include Gaussian symmetry and near-tie results, finite-sample convergence rates, and implications for sequential calibration. Contents of this deposit: Main research manuscript (21 pages). Technical supplement containing Appendices A–G (74 pages), including detailed mathematical proofs and convergence-rate analyses. Computational archive containing Python scripts, numerical verification material and reproduction instructions. Author: Shardul Mishra, Vellore Institute of Technology, India. Copyright: © 2026 Shardul Mishra. All rights reserved.