Feedback‐Driven Convergence, Competition, and Entanglement in Classical Stochastic Processes

ABSTRACT We present a dynamical theory of statistical convergence in which the law of large numbers arises from outcome‐outcome feedback rather than assumed independence. Defining the convergence field and its derivative, we show that empirical frequencies evolve through coupling, producing competition, finite‐ fluctuations, and classical entanglement. Using the Kramers‐Moyal expansion, we derive an Itô‐Langevin and Fokker‐Planck description, reducing in the symmetric regime to a time‐dependent Ornstein‐Uhlenbeck process. We propose variance‐based witnesses that detect outcome‐space entanglement in both binary sequences and coupled Brownian trajectories, and confirm entanglement through numerical experiments. Extending the formalism yields multi‐outcome feedback dynamics and finite‐time cross‐diffusion between Brownian particles. The results unify convergence, fluctuation, and entanglement as consequences of a single feedback‐driven stochastic principle.