A Stochastic Open-System Boundary Effective Field Theory for Late-Time Gravitational-Wave Relaxation

We construct a mathematically controlled, open-system boundary effective field theory (EFT) to model the late-time stochastic residual response in post-merger gravitational-wave spectroscopy. While deterministic strong-field merger dynamics remain strictly governed by the classical Einstein field equations, integrating out unresolved high-energy degrees of freedom at a timelike matching hypersurface exterior to the horizon generates a non-local boundary effective action. Using a linear spin-2 tensor sourcing framework where asymptotic gravitational perturbations couple to a primary boundary conformal field theory (bCFT) operator , we derive a non-local, dissipative mixed Robin boundary condition from variational stationarity of the coupled bulk-boundary action. We demonstrate a fundamental structural bifurcation within the induced self-energy loop integral: all ultraviolet-sensitive, regulator-dependent contributions are strictly polynomial and local in the time domain, allowing their systematic order-by-order absorption into a standard tower of local boundary counterterms . Conversely, analytic continuation of the infrared scale-invariant sector generates a universal, regulator-independent continuum response sector parameterized by the fundamental scaling dimension . Choosing the principal branch such that the non-meromorphic cut lies entirely along the negative imaginary frequency axis satisfies Titchmarsh’s theorem, guarantees causality in the upper half-plane, and ensures immunity to Ostrogradsky or runaway instabilities. Imposing simultaneous constraints from Kramers–Kronig contour convergence and spectral positivity establishes a derived conformal stability window 1.5 < ΔO < 3.5. Finally, utilizing the fluctuation-dissipation relation, we project the continuum spectral density into a structured non-local metric covariance operator Γ(ΔO, G_B,θ), The present framework should be interpreted as a phenomenological open-system EFT constrained by causality, renormalizability, and infrared scaling consistency.