History-Dependent Gravity XI: Gravitational Wave Phase Drift from Finite Memory Kernel as Cross-Epoch Verification of Emergent Time

This is the eleventh paper in the History-Dependent Gravity (HDG) series, providing the first falsifiable observational prediction of emergent time through gravitational wave phase drift detectable by LISA. The HDG framework unifies time, entropy, and cosmological evolution as emergent phenomena from a nonlocal gravitational memory kernel K(t,t'). We propose the Temporal Emergence Principle: physical time is defined as a monotonic functional of accumulated non-local correlations in the gravitational sector. A direct consequence is the modification of gravitational wave propagation. We derive three distinctive observational signatures that constitute a smoking gun for HDG: (i) a non-analytic kink in the frequency-dependent phase drift at f_kink ≃ 1/(2πτ), (ii) correlated non-exponential amplitude damping h(f) ∼ exp(−α f^γ) with γ ≠ 1, and (iii) log-normal spectral broadening. No alternative theory produces all three simultaneously. Crucially, the memory scale τ is not a free parameter but an RG invariant already determined from cosmological structure formation. This constitutes a cross-epoch consistency test: the same τ* must describe both galaxy clustering and gravitational wave propagation. Our Fisher forecast indicates that for τ* = 10^3 s and coupling ξ ≳ 3.5×10^−14, the kink is detectable at >5σ with LISA; for the fiducial value ξ = 10^−12, the naive significance exceeds 1000σ. A more realistic estimate accounting for waveform systematics and detector calibration reduces this by an order of magnitude, but the 5σ threshold is still reached for ξ ≳ 10^−13. A detection would provide the first experimental evidence that time itself is an emergent phenomenon.