Time Reversal, Entropy Relativity, and Computational Naturalness: A Unified Perspective on the Arrow of Time

We investigate time-reversal symmetry and its effective breakdown in complex dynamical systems. Although microscopic equations admit reversible trajectories, macroscopic reversal becomes operationally unattainable once a system exports correlations and builds structured, record-bearing organization. We formalize this via \emph{computational naturalness}: forward prediction typically requires only local causal data within a compact \emph{calculation cone}, whereas faithful reversal requires specifying a vastly larger \emph{retrodiction sphere} containing dispersed correlations (radiation, scattering records, entanglement, and environmental imprint). This asymmetry is quantified by a \emph{naturalness ratio} $\kappa = I_{\mathrm{rev}}/I_{\mathrm{fwd}}$, comparing the informational (Kolmogorov-optimal) description lengths needed for reversal versus prediction; for structured systems $\kappa\gg 1$ grows rapidly with complexity, sharply limiting any practical reversibility window. Concurrently, we introduce an explicit measure of \emph{entropy relativity}: an observer-anchored entropy defined as a Kullback--Leibler divergence on a causal hypersurface, reproducing Gibbs/von Neumann entropy up to an additive constant when equilibrium is used as reference while capturing the informational weight of nonlocal correlations relative to an observer's accessible prior. Applied to cosmology, this framework reframes the ``low-entropy initial condition'' problem: an early state that is near-maximal relative to its comoving coarse-graining and causal domain can appear low when compared against a later, expanded reference state with different accessible degrees of freedom, especially once horizon-scale correlations are accounted for. We classify systems into reversible, computationally fragile, and structurally irreversible regimes, and clarify the role of $CPT$ as the universally well-defined reversal mapping in relativistic quantum theory even when $T$ symmetry alone is violated. Together, these tools connect microscopic reversibility to macroscopic irreversibility through causal accessibility, exported correlations, and the computational asymmetry between prediction and reversal. We further show that thermal equilibrium is a timeless limit for \emph{operational} time: using an arrow functional defined as a Kullback--Leibler divergence between forward and appropriately reversed path measures, equilibrium is the unique macrostate with vanishing arrow (a stationary coarse-grained path measure). Away from equilibrium, an operational time direction \emph{emerges} from the stable causal updating of exported correlations and records. In that regime, macroscopic ``pure reversal'' would require an instantaneous state-map that is obstructed both mechanically (impulse idealization) and informationally (dispersed correlations), rendering reverse histories measure-suppressed.