We investigate the fundamental nature of time-reversal symmetry and its progressive breakdown in complex, structured dynamical systems. Drawing on the Fluctuation Theorem and recent advances in quantum time-reversal experiments, we reveal that the process of computational evolution—by which systems develop increasingly sophisticated mechanisms to resist entropy—appears to naturally enforce intrinsic non-time-reversible behavior. In parallel, we introduce a novel classification scheme that categorizes physical, biological, and computational systems according to their time-reversal properties. Our analysis suggests that devices such as electrochemical batteries, biological neural networks, and both natural and artificial computational architectures exhibit irreversibility not solely due to statistical entropy production but also as a consequence of deeply evolved structural and causal dependencies. Central to our framework is the concept of time-entropy momentum—a quantitative measure capturing the resistance of a system’s temporal evolution to entropy increase. We further hypothesize that this momentum imposes a robust bound on the duration over which a system can sustain time-reversible dynamics before irreversible entropy accumulation sets in. In low-entropy, highly organized structures—especially those characteristic of the mature universe—this bound becomes vanishingly small, contributing to the observed unidirectional flow of time. Our findings thus lay a foundation for understanding the transition from microscopic reversibility to macroscopic irreversibility and offer new insights into the interplay between entropy, computational evolution, and the arrow of time.