Gravitational Memory and the Emergence of Time: Decoherence, Irreversibility, and Entropy in Nonlocal Quantum Gravity

In the twentieth paper of the History-Dependent Gravity (HDG) series, we develop a covariant effective field theory of gravity with temporal nonlocality, in which the metric dynamics depends on its past history through a causal memory kernel. The nonlocal structure is characterized by a logarithmic operator K () = (+ m₌₄₌ℂ²), introducing a physical mass scale m₌₄₌ that controls the finite relaxation time of gravitational memory. Using a Hubbard--Stratonovich transformation, we demonstrate that the nonlocal theory is dynamically equivalent to General Relativity coupled to an auxiliary scalar field---the memory field---with a gapped spectrum. This representation reveals that gravity, at the effective level, behaves as an open quantum system interacting with an internal reservoir of memory modes. We show that decoherence and dissipation arise only above a kinematic threshold p² > m₌₄₌², where real memory modes can be excited. Below this scale, the dynamics remains approximately unitary. This establishes decoherence as an emergent, scale-dependent phenomenon rather than a fundamental property of quantum gravity. We further demonstrate that the arrow of time emerges dynamically from the spectral asymmetry of the memory kernel and the selection of retarded boundary conditions. While the underlying theory is time-reversal invariant, irreversibility appears at the level of the effective quantum dynamics. Black hole entropy is shown to arise from the entanglement of memory modes across the horizon, leading universally to an area law S A/G, with subleading corrections controlled by the memory scale. This provides a unified interpretation of horizon thermodynamics in terms of gravitational information storage. The framework makes concrete, testable predictions, including frequency-dependent dispersion of gravitational waves, echo signals from black hole ringdown, modifications to the Newtonian potential, and scale-dependent suppression of primordial tensor modes. These results suggest that gravitational memory provides a minimal and universal mechanism linking quantum dynamics, irreversibility, and holography within a consistent effective field theory of gravity.