Irreversibility of recursive Heaviside memory functions: a distributional perspective on structural cognition

Modern AI systems excel at pattern recognition and task execution, but they often fall short of replicating the layered, self-referential structure of human thought that unfolds over time. In this paper, we present a mathematically grounded and conceptually simple framework based on smoothed step functions—sigmoid approximations of Heaviside functions—to model the recursive development of mental activity. Each cognitive layer becomes active at a specific temporal threshold, with the abruptness or gradualness of activation governed by an impressiveness parameter sᵢ , which we interpret as a measure of emotional salience or situational impact. Small values of sᵢ represent intense or traumatic experiences, producing sharp and impulsive responses, while large values correspond to persistent background stress, yielding slow but sustained cognitive activation. We formulate the recursive dynamics of these cognitive layers and demonstrate how they give rise to layered cognition, time-based attention, and adaptive memory reinforcement. Unlike conventional memory models, our approach captures thoughts and recall events through a recursive, impressiveness-sensitive pathway, leading to context-dependent memory traces. This recursive structure offers a new perspective on how awareness and memory evolve over time, and provides a promising foundation for designing artificial systems capable of simulating recursive, temporally grounded consciousness.