History-Dependent Gravity XIV: Temporal Kernel Geometry and Logarithmic Memory Signatures

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^ (-13/3) 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).