Uniqueness of the Canonical Half-Iterate

Abstract

A companion note (Existence of a Half-Iterate of the Exponential Map) stated, withoutproof, the Trappmann–Kouznetsov criterion singling out a canonical holomorphic half-iterateof exp, and remarked that its uniqueness half has been independently re-derived by twostructurally different arguments. This note supplies both in full. The first shows that thedeviation between any two candidate solutions is an entire, period-one function controlledentirely by its behaviour near the underlying fixed point, and pins that behaviour down by asharp growth threshold. The second follows Trappmann and Kouznetsov’s own route — thatthe deviation is itself invertible, hence affine, hence trivial — but closes a gap their publishedargument leaves classical: we prove from scratch, with no citation to Casorati–Weierstrassor the classification of isolated singularities, that an entire function with an entire inversemust be affine. Both routes reduce uniqueness entirely to the single remaining open problemalready identified in the companion note: exhibiting one concrete solution satisfying thecriterion’s hypotheses.