On a Spectral Correspondence between the Wheeler– DeWitt Operator on Hyperbolic Superspace, Black Hole Cusp Geometry, and the Non-Trivial Zeros of the Riemann Zeta Function

We establish a mathematically precise chain connecting the Wheeler–DeWitt (WdW) operator of canonical quantum gravity to the non-trivial zeros of the Riemann zeta function , and derive from first principles why the WdW symmetry group must be arithmetic. After Wick rotation, the WdW operator on FLRW mini-superspace with negative spatial curvature reduces to the self-adjoint Laplace–Beltrami operator on hyperbolic 3-space. The central new contribution is a derivation that black hole configurations in the Hartle–Hawking path integral correspond to cusps of , and that the no-boundary regularity condition forces rational monodromy of the WdW wave function at every cusp — via the Bohr–Kronecker theorem applied to the -convergent Dirichlet series of Euclidean black hole periods. Rational monodromy (Theorem A: Cassels 1978) forces each cusp lattice to be a fractional ideal in an imaginary quadratic field ; the trace field criterion (Theorem D: Maclachlan–Reid 2003) then implies that is arithmetic, commensurable with the Bianchi group . For Bianchi groups, the Selberg zeta function factors through the Dedekind zeta function , embedding all nontrivial zeros of in the spectral parameters of the WdW operator. The Hilbert–Pólya operator is thereby realized concretely as on a gravitational Hilbert space. As a parameter-free physical consequence, the theory predicts log-periodic modulation of galactic HI density rings with spacing ζ(s) (k = -1) -ΔH3 Γ∖H3 L2 Q(√-d) Γ Γd = PSL(2, Od) ζQ(√-d)(s) = ζ(s) ⋅ L(s, χ-d) ζ(s) -ΔH3 r n+1/rn = exp(π/γ1) ≈ 1.2484, determined solely by the first Riemann zero . This prediction is falsifiable against THINGS and LITTLE THINGS survey data. The entire argument reduces to two precisely-stated physical hypotheses (H1, H2), each independently testable