An Exactly Solvable Alternating Pole Tower: Closed-Form Resurgence, Lambert W Asymptotics, and Perturbative Zeta Degeneracies

Abstract

We present the comprehensive asymptotic and resurgent analysis of an alternating pole tower function characterized by non-isolated essential singularities. Unlike generic asymptotic models where exponential data must be numerically fitted, this system is exactly solvable simultaneously on both sides of the resurgence relation. By applying Poisson resummation to the alternating lattice sum, the resulting saddle-point equations are solved in closed form in terms of the Lambert W function, providing exact analytical expressions for all transseries coefficients, Stokes constants, and dominance-exchange rays. Concurrently, the endpoint (perturbative) sector is summed exactly, yielding a factorially divergent rational sequence governed by Bernoulli numbers. We detail an algorithmic framework ("Stirling Gates") for generating high-order branch series and introduce a one-parameter deformation family ($S_\alpha$) that connects the system to the Riemann Zeta function. Remarkably, we show that the odd zeta values, including Apéry's constant ($\zeta(3)$), emerge as a purely perturbative datum through the transversality of the system's boundary degeneracies. All analytical formulas are corroborated by ultra-high-precision numerical experiments (up to 230 digits), providing a rigorous benchmark for modern exponential asymptotics and resurgence theory.