Orbit Regularization: The Hurwitz Zeta Function in the Modulo-30 Statistical Sieve and the Discrete Spectrum of Prime Distribution

This paper proposes a fundamentally new theoretical framework—Orbit Regularization—that establishes a systematic and precise correspondence between the eight orbits of the modulo-30 statistical sieve and special values of the Hurwitz zeta function at rational parameters. I prove that for every orbit r∈1, 7, 11, 13, 17, 19, 23, 29, the reciprocal square sum over all integers greater than 1 on that orbit converges precisely to the Hurwitz zeta function value ζ (2, r/30) /900. At the N=10¹0 scale, the absolute deviations between the measured values and the theoretical limits for all eight orbits lie between 9. 39E-12 and 3. 33E-9—all within the accumulated rounding-error limit of double-precision floating-point arithmetic. Each orbit independently exhibits a "freezing" phenomenon. Furthermore, I have observed in quantum simulators that the prime factors of composite numbers on the eight orbits strictly obey the phase and period structure, providing direct physical verification for orbit regularization. I draw a structural analogy between this discrete number-theoretic framework and the Zeta regularization technique employed in the Casimir effect of quantum field theory, and discuss its deep resonance with the Hilbert–Pólya conjecture.