The Emergence of Prime Distribution from Low-Dimensional Deterministic Chaos

The distribution of prime numbers is a cornerstone of mathematics, yet its underlying origin remains one of science's deepest mysteries. While probabilistic models successfully describe the asymptotic density of primes, they inherently fail to explain the arithmetic rigidities and short-range correlations that define the sequence. Here we show that prime statistics emerge naturally from a low-dimensional deterministic physical system. By modeling the prime sieve as a non-autonomous chaotic process subject to dissipation, we identify a topological isomorphism between the arithmetic constraints of primes and the symbolic dynamics of the Logistic map at the band-merging point. Our model quantitatively reproduces the Twin Prime Constant (C2≈0.6602) with high precision, governed by a decay exponent strictly constrained by number-theoretic axioms. This finding suggests that arithmetic randomness is not a form of noise, but a signature of weak chaos, offering a unified physical framework that bridges the discrete world of number theory and the continuous world of nonlinear dynamics.