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.