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 $\Psi$ to the Weyl tensor squared, $S \supset \int d^4x \sqrt{-g} \, \xi \Psi C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$, we derive the modified gravitational wave equations and show that the memory kernel emerges naturally as the retarded Green's function of the $\Psi$ field. This induces a frequency-dependent phase drift $\delta\phi(f)$ that transitions from $\propto f$ at low frequencies to $\propto f^{-1}$ at high frequencies, creating a sharp kink at $f_* = (2\pi\tau_*)^{-1}$. Crucially, the memory scale $\tau_*$ 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 $\tau_*$ 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.