**Version 1. 1 — Corrected phase drift scaling** This version corrects the gravitational-wave phase drift formula from δΨ (f) ∝ f^-13/3 log f to δΨ (f) ∝ f^-5/3 log f, following referee feedback. The correction does not affect the main conclusions: the effect remains functionally orthogonal to standard PN parameters and GR tail effects, and is in fact enhanced since it now appears at leading order rather than being suppressed. All other results (kernel derivation, Fisher analysis, detectability forecasts) remain valid with updated exponents. This is Paper XIV in the History-Dependent Gravity (HDG) series. Building on the dynamical relaxation framework of HDG XIII, we derive the observational consequences of the scale-free temporal kernel K (t) ∝ 1/t that emerges from the relaxation dynamics. Main results: (1) The kernel is uniquely selected by scale invariance of the relaxation equation (Appendix C). (2) Fourier transform yields a universal logarithmic response K̃ (ω) ∼ -log (iω) (Appendix A). (3) Two distinct observable signatures: - Source-level: GW phase drift δΨ (f) ∝ f^ (-5/3) log f in compact binary inspirals (Section 3) - Propagation-level: logarithmic dispersion during cosmological travel (Section 4) (4) Fisher matrix analysis shows LISA can probe the fundamental coupling λ₀ down to ~10⁻³ through hierarchical population analysis (Appendix B). The logarithmic phase drift is irreducible — it cannot be absorbed into standard waveform parameters (masses, spins, distance) — and provides a direct, falsifiable test of directional time geometry. All observable effects are controlled by the same fundamental parameter λ₀, with λₑff encoding its history-dependent realization. Part of the series "Temporal Nonlocality and Fundamental Physics" (Zenodo community).
Alik Gimranov (Sat,) studied this question.