Geometry of Temporal Memory: History-Dependent Gravity as an Emergent Extra Dimension

We show that the temporal nonlocality inherent in History-Dependent Gravity (HDG) can be geometrised by introducing an auxiliary memory coordinate τ. The memory kernel K(t-t') emerges as the boundary-to-boundary propagator of a local bulk field in a (1+1)-dimensional extended spacetime. This reformulation restores locality, provides a natural open-quantum-system interpretation (Nakajima-Zwanzig equation), and yields a universal correction to black hole entropy: ΔS = A/(4G₀) K̃(0). Furthermore, the memory-induced geometry modifies spatial correlation functions, replacing the exponential Yukawa decay with power-law tails when the kernel exhibits long-range behaviour (K(t) ∝ t^(-β)). For β=1/2, we find G(r) ∝ r^(-3/2) in the infrared. These effects are observable as frequency-dependent phase shifts and damping in gravitational waves (|Δφ| ∝ f^(β-1) L, Γ ∝ f^(β-1)), modifications of quasinormal mode spectra, and scale-dependent running of the primordial spectrum, providing falsifiable tests for next-generation detectors and cosmological surveys. We also discuss how strong gravitational fields (e.g., near black hole horizons) renormalise the memory kernel, leading to enhanced long-term correlations rather than information loss. A natural horizon cutoff regularises the zero-mode divergence, yielding finite entropy corrections.