Recursive Time Dynamics: A Unified Mathematical Framework for Subjective Time Across Biological and Artificial Systems

Abstract

Time perception varies dramatically across individuals, species, and cognitive states, yet existing models-Scalar Expectancy Theory, Striatal Beat Frequency, and Drift-Diffusion frameworks-fail to explain this diversity through unified mechanisms. We propose the Recursive Time Dynamics Model (RTDM), a mathematical framework positing that subjective time emerges from three interacting components: (1) recursion speed ω(t), representing neural or computational update rates; (2) coherence C(t), quantifying synchrony across system layers; (3) periodic rhythms S(t), capturing circadian and metabolic cycles. The model defines subjective time as τ(t) = ∫ k•ω(s)•(1+λC(s))ds over finite integration windows, predicting that faster recursion and higher coherence accelerate time perception, while circadian modulation introduces predictable diurnal variations. Critically, RTDM extends beyond human cognition to animal metabolic time-scaling with body mass M-1/4-and artificial systems, where iteration rates and distributed coherence generate analogous temporal metrics.