Gravitational Wave Phase Drift from a Non-Local Memory Kernel: A Covariant Extension of General Relativity

We construct a minimal, covariant, and ghost-free modification of General Relativity that predicts a unique, non-analytic ``kink'' in the gravitational wave phase evolution. Starting from an effective action coupling a scalar field Ψ to the Weyl tensor squared, S ⊃ ∫ d⁴x √-g \, ξ Ψ CμνρσCμνρσ, we derive the modified gravitational wave equations and show that the memory kernel emerges naturally as the retarded Green's function of the Ψ field. This induces a frequency-dependent phase drift δφ(f) that transitions from ∝ f at low frequencies to ∝ f⁻¹ at high frequencies, creating a sharp kink at f_* = (2πτ_*)⁻¹. Crucially, the memory scale τ_* is not a free parameter but an RG-invariant already determined from cosmological structure formation, providing a cross-epoch consistency test between galaxy clustering and gravitational wave propagation. We perform a Fisher forecast for LISA, demonstrating that the kink is detectable at high significance, with the scale τ_* measurable to sub-percent precision. These results provide a concrete observational test of non-local gravitational dynamics and establish gravitational wave phase measurements as a probe of temporal non-locality in effective theories of gravity.