Asymptotic Solutions of the Tetration Equation

The paper states and sketches a proof of the Shell–Thron result (existence of a holomorphic inverse Abel function F_λ with F_λ(s+1)=e^{μ F_λ(s)} and period 2πi/λ for Re λ > −log|μω_μ|) via the β-construction, linearization, and a pull-back/implicit-function argument, but several key steps are only heuristically justified (e.g., reliance on an unspelled “Inverse Abel Theorem,” measure-zero assertions, and normal convergence) . The model’s proof idea (back-iterating with a fixed local inverse T and defining X_n(s)=T^n(β(s+n))) is elegant but has a serious domain-of-definition gap: T is defined on f(U) and cannot be iterated n times on arbitrary points of U unless one shows β(s+n)∈f^n(U), which is not established; derivative bounds for T^n on U are likewise unjustified. Thus, both arguments need substantial repairs. The central claims are plausible, but the provided proofs are incomplete.