A Schur-Convex Entropy Increment for the Self-Power Escort Transformation

We study the Shannon entropy increment generated by the self-power escort transformation on a finite probability simplex, in which each coordinate is raised to its own value and the resulting weights are normalized. With the continuous boundary convention 0⁰ = 1, we prove that entropy increases strictly at every nonuniform probability vector, in every dimension. The proof establishes the stronger fact that the entropy increment is strictly Schur-convex in the interior. Its main ingredient is a monotonicity estimate for the coordinate derivatives, based on an aggregation inequality for self-powers and a uniform two-point weight bound. A facewise argument extends strict entropy increase to all boundary points. We also prove binary output majorization and construct interior counterexamples to output majorization in every dimension at least five. Finally, we determine the sharp quadratic entropy-production coefficient near the uniform distribution. All map-specific ingredients are derived here; the entropy theorem does not assume a separate convergence or contraction theorem for the transformation.