Existence of a Half-Iterate of the Exponential Map

We consider the classical problem of a compositional square root of the exponentialmap: a function φ with φ ◦ φ = exp. We first give a short, entirely elementary proof that astrictly increasing solution exists on R, using nothing beyond single-variable calculus. Suchsolutions are not unique, and the elementary one is not smooth in the usual sense. Wethen turn to the much harder problem Kneser originally posed: a solution that is genuinelyholomorphic, singled out uniquely among such solutions by a translation-covering criteriondue to Trappmann and Kouznetsov. We state that criterion precisely, prove a generalcovering lemma it depends on, and report the current state of the existence question for ithonestly: uniqueness is fully established; existence is reduced to two precisely stated analyticobligations that remain open.