The Limits of a Family; of Asymptotic Solutions to The Tetration Equation

The paper states a construction of a holomorphic tetration tetβ on C\(−∞,−2] with tetβ(s+1)=e^{tetβ(s)}, tetβ(0)=1, and a real-analytic bijection on (−2,∞) (Theorem 7.3), built from the infinite composition β(s) and a diagonal iterated-log limit (Fn(s)=log^∘n β(s+n)), after selecting a variable λ(s)=1/√(1+s) to avoid singularities. However, crucial steps are only sketched: the ‘Desired Mapping Theorem’ for λ(s), uniform branch control for the iterated principal logs, uniform domains for contraction estimates in Theorem 5.1 and Theorem 6.1, and especially the ‘Non-real Lemma’ and Non-zero Theorem used to rule out extra singularities in the upper half-plane rely on informal “looks like” arguments without rigorous justification . The candidate (model) solution gives a sharper telescoping identity and a plausible matched-asymptotic O(n^{-1/2}) control to prove convergence and the functional equation, plus a clearer monotonicity argument on (−2,∞). But it also leaves nontrivial estimates (tail asymptotics T_n(u), sector/branch control for principal logs, and normalization-by-shift) at the level of sketches. Consequently, both the paper and the model solution are promising but incomplete.