Gravitational Origin of the Hilbert–Pólya Operator: Wheeler–DeWitt Arithmeticity, Black Hole Cusps, and the Non-Trivial Zeros of the Riemann Zeta Function

Abstract. We derive, from the Einstein–Hilbert action and the Hartle–Hawking no-boundary proposal, that the Wheeler–DeWitt (WdW) symmetry group of canonical quantum gravity is necessarily an arithmetic Kleinian group commensurable with a Bianchi group Γd=PSL (2, Od) Γd=PSL (2, Od). The derivation proceeds through five rigorously established steps: (1) the Euclidean WdW operator on k=−1k=−1 FLRW superspace reduces to the Laplace–Beltrami operator −ΔH3−ΔH3; (2) Euclidean black hole configurations in the Hartle–Hawking path integral correspond to cusps of Γ3Γ3, via the Gibbons–Hawking saddle-point analysis and Thurston's boundary of Teichmüller space; (3) L2L2-normalizability of the wave function (Bekenstein–Hawking entropy suppression) forces rational monodromy via the Bohr–Kronecker theorem; (4) rational monodromy implies ideal-lattice cusps in Q (−d) Q (−d) (Cassels 1978) ; (5) the Maclachlan–Reid trace-field criterion (2003) implies arithmeticity, and the Elstrodt–Grunewald–Mennicke theorem (1998) identifies ΓΓ with ΓdΓd. For Bianchi groups, the scattering matrix of −ΔH3−ΔH3 factors through the Dedekind zeta function ζQ (−d) =ζ⋅L (⋅, χ−d) ζQ (−d) =ζ⋅L (⋅, χ−d), embedding all non-trivial zeros of ζ (s) ζ (s) among the WdW spectral resonances. The Hilbert–Pólya operator is thereby realized as −ΔH3−ΔH3 on a gravitational Hilbert space, and the Riemann Hypothesis is equivalent to a spectral gap λn≥14λn≥41 for this operator. As a parameter-free physical consequence, the theory predicts log-periodic galactic HI ring spacing rn+1/rn=exp⁡ (π/γ1) ≈1. 2484rn+1/rn=exp (π/γ1) ≈1. 2484, where γ1=14. 13472…γ1=14. 13472… is the first Riemann zero. This prediction is falsifiable against THINGS and LITTLE THINGS survey data.