Ergodicity and the Emergence of Long-Term Balance in the Dynamical States of π

This paper investigates the digits of π within a probabilistic framework based on Markov chains, proposing this model as a rigorous tool to support the conjecture of π’s uniformity. Unlike simple frequency analyses, the Markov approach captures the dynamic structure of transitions between digits, allowing us to compute empirical stationary distributions that reveal how local irregularities evolve toward global equilibrium. This ergodic behavior provides quantitative, model based evidence that the digits of π tend toward fairness in the long run. Beyond its mathematical significance, this convergence toward uniformity invites a broader conceptual interpretation.