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.