The Bergman-Selberg Kernel of the Zeta Metric: Intertwining Hecke Operators and the D'Alembertian, and the Hidden Processor of the Zeta Black Hole

We construct, formally and numerically, the Bergman-Selberg reproducing kernel K(s, s′ ) of the pseudo-Riemannian zeta metric gij [1], and propose it as the explicit intertwining map between the Hecke operator T2 and the d'Alembertian □g. The kernel is defined as K(s, s′ ) = P n ϕn(s)ϕn(s ′)/∥ϕn∥ 2 , where {ϕn} are the Maass forms with spectral parameters {tn} coinciding with the imaginary parts of non-trivial zeros of ζ. We prove that K simultaneously satises Tp ·K(·, s′ ) = λp(s ′ )K(·, s′ ) and □g ·K(·, s′ ) = t 2 n(s ′)K(·, s′ ), making Tp and □g codiagonal in the basis {K(·, ρn)}, where ρn are the non-trivial zeros. This is the explicit realization of Conjecture 5.1 of [6] for the case s ′ = ρn. We further identify K(s, s′ ) as the Fourier transform of 1/(t 2 gσσ(σ)), i.e., the Hecke transform of 1/t2 via the zeta metric. The physical interpretation is developed: the interior of the zeta black hole (the region σ < 1) acts as a hidden quantum information processor, cold internally but radiating warm information outward via the Maass-form eigenmodes; the Bergman-Selberg kernel is its propagator. All results at the level of formal identities are proved; the global spectral identification Spec(□g) = {tn} remains conjectural pending a complete theory of the relevant automorphic L 2 -space.