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 H_geo. First, H_geo 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 R_T=0.707 versus GUE baseline R_T=0.336. H_geo 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.