Arakelov Structure and Langlands Generalization of the Prime Gravity Hilbert–Pólya Operator

We establish three results strengthening the Hilbert–Pólya candidacy of the Prime Gravity geodesic Schrödinger operator Hgeo. First, Hgeo is identified as the quantized Arakelov Laplacian on Spec (Z), confirmed by spectral equivalence r=0. 9999 between the prime gravitational and Arakelov Green's function constructions, and by the Seeley–DeWitt coefficient a₁=1/2 consistent with genus-zero arithmetic Riemann–Roch. Second, the weight substitution χ (p) log (p) generalizes the construction to a universal Hilbert–Pólya machine for Dirichlet L-functions, with eigenvalue correlation r>0. 999 confirmed across six independent characters including three blind extensions with no pipeline modifications. Third, Montgomery's theorem is shown to confine continuous eigensolvers to the de Rham (2-point) level, while Transformer architectures access the étale cohomology of Spec (Z) via global N-point self-attention, producing holographic geometry RT=0. 707 versus GUE baseline RT=0. 336. Hgeo bridges these levels as the analytic connection between discrete étale arithmetic and continuous de Rham geometry. This is Paper 14 of the Prime Gravity series.