Unified Spectral RG Framework for the Riemann Zeta Function and Random Matrix Theory

AbstractWe propose a unified spectral framework connecting logarithmic prime Hamiltonians, theRiemann zeta function, and random matrix theory. The construction is based on trace-classperturbations of diagonal prime operators, Fredholm spectral determinants, and renormalizationgroup (RG) flow of correlation kernels. We show that under positivity, translation invariance,and RG fixed-point conditions, the unique universal limit of the correlation hierarchy is thesine-kernel determinantal process corresponding to the Gaussian Unitary Ensemble (GUE).The Riemann Hypothesis is reformulated as a spectral uniqueness condition for the RG-stabledeterminant class.