The Emergence of Prime Distribution from Low-Dimensional Deterministic Chaos

The distribution of prime numbers is one of the oldest and most fundamental problems in number theory. Since Riemann, the dominant paradigm has treated the occurrence of primes as a pseudo-random process, typically approximated by probabilistic models such as the Cramér model. However, these stochastic methods inherently fail to capture the rigid arithmetic constraints and short-range correlations that define the sequence of primes. This research report details a disruptive perspective: the statistical properties of prime numbers can naturally emerge from a non-autonomous, low-dimensional deterministic chaotic system. By placing the Logistic map at the band-merging point ($u \approx 1.5437$) under asymptotic density scaling, we successfully reproduced the Prime Number Theorem and the Hardy-Littlewood constant with high precision. Beyond macroscopic density, our dynamic framework also captures microscopic features that purely random models cannot explain: the discrete spectrum of forbidden gap sizes, the short-range anti-correlation of consecutive gaps, and the critical intermittency of twin prime events characterized by power-law waiting times. Furthermore, we calculated a positive but suppressed Lyapunov exponent ($\lambda \approx 0.1$), identifying the prime sequence as a "weakly chaotic" system distinct from both periodic order and white noise. Combined with the latest theories of arithmetic Schrödinger flows and quantum chaos research, these results suggest that the prime sequence belongs to a dynamic universality class operating at the edge of chaos, providing a deterministic physical basis for the apparent randomness of arithmetic structures.