On Certain Spectral Operators Associated with the Riemann Zeta Function - with consequences for the Riemann Hypothesis

This paper develops a spectral-operator framework for the Riemann zeta function, utilizing renormalized trace identities and distributional analysis to address the Hilbert-Pólya program. The authors construct a canonical "infinite-prime" self-adjoint operator through the renormalized limit of prime-weighted scaling perturbations. Unconditionally, the work establishes that the renormalized spectral distribution of this operator coincides with the nontrivial zero distribution of the Riemann zeta function. The study identifies a structural low-frequency obstruction that prevents the associated Prime Renormalized Spectral Form (PRSF) from being coercive on the full Hilbert space. Consequently, the Riemann Hypothesis is reformulated as a rigidity principle: it is equivalent to the existence of a unique, canonical admissible boundary condition that restores coercivity to the PRSF.