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 H_MS on the TIC/CIT modular surface M = SL(2,Z)\H^2, with real spectral parameters E_rho = H^2(tau^2 + 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 dS_2 ≅ H^2; (II) the Gibbons-Hawking thermal periodicity and sigma-field S-duality determine the modular surface M_TIC = SL(2,Z)\H^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 M_TIC 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.