Domain of Validity of the Reheating-Imprint Kernel for Axions: Controlled Late-Time Asymptotics from Retarded Green Functions

This work establishes a controlled late-time asymptotic structure for reheating-imprinted axion initial data based on a retarded Green-function formulation. Under a damped-oscillatory source and a slowly varying reheating background, the sourced response admits a decomposition into a local amplitude term and a phase-sensitive memory sector governed by the initial-boundary propagation kernel G(u,0)G(u,0)G(u,0). The kernel fixes the universal oscillatory-decay structure, while non-universal information is encoded in a matching coefficient K(Ω,w)K(\Omega,w)K(Ω,w) and a subleading frequency correction δω(u;Ω,w)\delta\omega(u;\Omega,w)δω(u;Ω,w), with controlled relative error of order O(ω−1)O(\omega^{-1})O(ω−1). A key result is the isolation of a memory diagnostic variable ZmemZ_{\rm mem}Zmem, which directly tracks the kernel-controlled phase. This provides a clear separation between universal asymptotic structure and model-dependent matching data, enabling robust interpretation of phase-sensitive reheating imprints without extending the kernel beyond its regime of validity.