The Spectral Correspondence as Theorem: Hyperbolic Geometry, Eisenstein Series, and the Zeros of the Riemann Zeta Function

We establish the Spectral Correspondence Hypothesis stated in TIC/CIT Paper 7: the non-trivial zeros of the Riemann zeta function zeta (s) appear as scattering resonances of the Mukhanov-Sasaki Hamiltonian HMS on the TIC/CIT modular surface M = SL (2, Z) ², with real spectral parameters Eᵣho = H² (tau² + 1/4) for zeros rho = 1/2 + i*tau on the critical line. The proof rests on three pillars: (I) the de Sitter MS equation is identified with the eigenvalue problem of the Laplace-Beltrami operator on the Poincare upper half-plane via the conformal isomorphism dS₂ ≅ H²; (II) the Gibbons-Hawking thermal periodicity and sigma-field S-duality determine the modular surface MTIC = SL (2, Z) ²; (III) the Eisenstein series scattering determinant phi (s) = xi (2s-1) /xi (2s) has poles at s = rho/2 for every non-trivial zero rho of zeta (s), producing resonant states with real spectral parameter iff rho lies on the critical line. The Selberg trace formula for MTIC is shown to be equivalent to the Riemann-Weil explicit formula, with prime geodesics corresponding to prime numbers. Combined with Paper 7, this gives the Riemann Hypothesis as a corollary. Paper 8 of the TIC/CIT series.