Discrete Dynamical System via the Mod 30 Statistical Sieve and Its Spectral Correspondence with the Zeros of the Riemann ζ Function

This paper reports a discrete dynamical system based on the mod 30 statis- tical sieve, whose spontaneously generated eigenvalue spectrum Pk exhibits a systematic, cross-scale exact correspondence with the imaginary parts γn of the non-trivial zeros of the Riemann ζ function. At two independent scales, N = 10⁶ and N = 10⁷, the Pearson correlation coefficients between Pk and γn are 0. 999704 and 0. 999945, respectively, with average relative errors of 0. 3310% and 0. 3221%. The normalized spacing distributions of both sets are highly consistent with the GUE theoretical predictions of Montgomery’s pair correlation function. I identify the algebraic origin of this “jump-type selec- tion rule”: the sieve employs cross-iterative moduli Mn = qn × qn+1, which produce a selection rule in discrete frequency space that picks out only the most stable subset of zeros under GUE statistics. The sieve has been mapped to quantum circuits on quantum simulators (pyQPanda and Qiskit), and γ15 is captured with a relative error of 0. 0712%. This discovery provides a com- putable discrete model for the Hilbert–Pólya conjecture based on elementary number theory.