Cycles of Disequilibrium in the Digits of π: Combinatorial Modeling of a Non-Random Local Equilibrium

We identify a local regularity in the digits of π: prolonged absences (gaps) of a given digit are followed by temporary overrepresentations (catch-ups, 12 %–30 %), before returning to an average frequency of 10 % over larger windows. This dynamic reveals a non-random structure, where π satisfies the law of large numbers globally but violates the i.i.d. hypothesis locally. We propose a combinatorial feedback model, based on a weighted cumulative deficit and a regulation coefficient k ≈ 0.05, introducing the disequilibrium–re-equilibration cycle as a unit of analysis. This approach suggests an internal combinatorial regulation in π, guiding occurrences towards equilibrium in the medium term.