Spectral Isomorphism between Renormalization Flow in Non-Autonomous Quadratic Maps and Riemann Zeros

The non-trivial zeros of the Riemann ζ function are widely conjectured to correspond to the eigenvalue spectrum of an unknown quantum chaotic system (the Hilbert-Pólya conjecture). While Random Matrix Theory (RMT) successfully describes the local statistical correlations of these zeros (GUE statistics), finding a deterministic dynamic operator that reproduces the global asymptotic behavior of the zeros remains an open challenge. In this work, we propose a dynamical model based on a Non-autonomous Logistic Map. By introducing a physically constrained renormalization flow k(n) ~ 1/ln n to drive the system's control parameter, we break classical fractal self-similarity in phase space. Utilizing massive parallel computation (256 cores), we generated the first 10⁴ eigenmodes of this operator. Phase Unwrapping analysis reveals that the cumulative phase of the dynamic system exhibits a strict global linear isomorphism with the Riemann zeros, achieving a correlation coefficient of R² > 0.997. Although a linear scaling deviation of ~ 4% persists, residual analysis uncovers smooth, coherent fluctuations, suggesting the existence of high-order perturbative terms in the Hamiltonian. This result provides a concrete physical pathway to reconstruct the Riemann ζ function via non-autonomous chaotic systems.