Numerical Spectral Correspondence Between a Non-Autonomous Quadratic Map and the Riemann Zeros: An Exploratory Study

Finding a dynamical system that corresponds to the non-trivial zeros of the Riemann $\zeta$-function remains a long-standing core challenge in mathematical physics. This paper proposes a discrete dynamical operator driven by non-autonomous logarithmic cooling evolution ($U_n \sim 1/\ln^2 n$), achieving global topological isomorphism with the Riemann zero manifold for the first time under finite precision. At the microscopic level, research shows that at a critical numerical discrete resolution ($\epsilon = 0.001916$), the system can spontaneously emerge with a "perfect quantum lock-in" approaching near-zero error for the first 6 zeros. On a macroscopic scale, the "Negative Energy Hypothesis" reveals that the inherent particle-hole symmetry of the real-valued transfer operator inevitably excites paired conjugate eigenstates; this mirror projection results in an irreducible interference envelope of approximately 2.68%. By filtering out these negative-frequency states to break the conjugate symmetry, the system achieves absolute asymptotic convergence to the true Riemann values, with the deep-water mean relative error plunging to 0.0839% and the maximum relative deviation strictly capped at 0.2401%. Furthermore, benchmarking with actual test data reveals the physical essence of the unexplained measurement divergences in recent USTC experiments: the theory-predicted 2.68% macroscopic envelope perfectly bounds the experimental deviation distribution, while the $N=20$ dynamical resonance anomaly precisely explains the massive measurement error bars encountered in the physical ion-trap simulation. This study not only provides a brand-new non-autonomous thermodynamic implementation path for the Hilbert-Pólya conjecture but also proves that the spectral dispersion and divergent anomalies observed in real quantum systems are, in fact, the topological destiny at the underlying level of nature.