TOWARD A SPECTRAL PROOF OF THE RIEMANN HYPOTHESIS: SPECTRAL UNITARITY, HECKE ALGEBRAS, AND AUTOMORPHIC SCATTERING IN ADELIC LANGLANDS MANIFOLDS

Abstract. We present a unified spectral framework intended to provide a possible route
toward a proof of the Riemann Hypothesis. By embedding the completed Riemann zeta
function into the automorphic spectral geometry of the adelic quotient space, we develop
a chain of constructions linking automorphic scattering, Hilbert–P´olya-type operators, and
Eisenstein series. The resulting framework identifies several key mathematical statements
whose establishment would imply the Riemann Hypothesis. We prove a number of foundational results, formulate the remaining critical steps explicitly, and discuss computational
architectures capable of validating the proposed program.