The Emergence of Prime Distribution from Low-Dimensional Deterministic Chaos

The distribution of prime numbers is a cornerstone of mathematics, yet its generative origin remains elusive. While probabilistic models describe the asymptotic density of primes, they fail to explain the arithmetic rigidities and short-range correlations that define the sequence. Here, we propose a deterministic dynamical framework that models the prime sieve not as a stochastic process, but as a non-autonomous chaotic system subject to dissipation. By identifying a topological isomorphism between the arithmetic sieve and the symbolic dynamics of the Logistic map at the band-merging point ($u_c \approx 1.5437$), we reveal that the "randomness" of primes is a signature of weak chaos. Crucially, we demonstrate that the famous Twin Prime Constant ($C_2 \approx 0.66016$)—traditionally derived from probabilistic circle methods—emerges naturally as a fixed point of our dynamical system, with numerical precision exceeding $10^{-4}$ and no manual parameter tuning. This work suggests that the distribution of primes is the trajectory of a deterministic system evolving at the "Edge of Chaos," offering a novel heuristic pathway to bridge Number Theory and Nonlinear Dynamics.