Information-Theoretic Origin of Time's Arrow: Correlation Asymmetry from Finite Memory Capacity

The arrow of time is traditionally attributed to the second law of thermodynamics and the low-entropy initial condition of the universe. Here we propose that the arrow emerges from a more fundamental property: the finite memory capacity of matter, which forces irreversible information erasure in its internal degrees of freedom. We extend stochastic thermodynamics by introducing a symbolic entropy sector that tracks logically irreversible information erasure in finite-memory systems. Under standard Markovian assumptions with a Crooks-type fluctuation relation extended to the symbolic sector, we show that time correlation functions become asymmetric whenever the system has a nonzero forgetting rate Ξcr > 0. Beyond Markovianity, we derive a lower bound linking directional correlation currents to symbolic entropy production, structurally analogous to thermodynamic uncertainty relations. The framework yields a sharp decoupling prediction: thermodynamic entropy production and symbolic forgetting are independent quantities. Systems with large thermodynamic entropy production but negligible forgetting (e. g. , plasma turbulence, fast quantum scramblers) exhibit time-symmetric symbolic correlations; systems with significant forgetting but arbitrarily small thermodynamic entropy production (e. g. , slowly aging glasses, near-Landauer information engines) exhibit a symbolic arrow without a thermodynamic one. This decoupling is the primary empirical discriminator between the proposed framework and standard thermodynamics. Idealized free photons and empty space, lacking metastable symbolic memory registers, exhibit no symbolic arrow. Matter, by contrast, inevitably forgets, casting the shadow we call the arrow of time. Time itself remains neutral; the direction we experience arises from how matter stores and forgets information.