A Prime-number Based Computational Toy Model for Irreversible Time and Information Growth

The nature of time and the growth of information remain foundational problems in physics and computation. Standard information-theoretic frameworks, such as Shannon entropy, primarily address the transmission of information and are largely insensitive to history, computational cost, and irreversibility. In this paper, we introduce a conceptual toy model—referred to as the Prime Universe—in which temporal progression is defined by the irreversible realization of successive prime numbers generated by a deterministic algorithm. Although the governing rule is simple, the realized history accumulates irreducible computational structure. This model illustrates how an arrow of time and monotonic information growth can emerge from irreversible computation without invoking randomness or thermodynamic assumptions. The model is not proposed as a description of physical reality, but as a minimal thought experiment clarifying the distinction between description, realization, and computational history.