The Geometric and Spectral Origin of the Riemann Hypothesis: The 30° Resonance Triangle, Wheeler–DeWitt Operator, and Spectral Gap Equivalence

We present a unified theoretical framework connecting the Riemann Hypothesis to spectral theory on arithmetic hyperbolic three-manifolds, motivated by canonical quantum gravity. Our central geometric observation is that the functional equation of ζ (s) (s) ζ (s) forces every non-trivial zero ρ=σ+iγ = + i ρ=σ+iγ to participate in a conjugate wave pair (ψρ, ψρ∗) (_, ^*) (ψρ, ψρ∗), and the stability condition for this pair — equal efficiency triangles — is equivalent to σ=12 = 12 σ=21, geometrically realized as a triangle with apex angle 30°30° 30°. Via the Selberg–Dedekind factorization theorem for Bianchi groups Γd=PSL (2, Od) d = PSL (2, Od) Γd=PSL (2, Od), we establish a precise equivalence: RH ⟺ λn≥14 for all scattering resonances of −ΔH3 on Γd3RH \;\; ₙ 14 for all scattering resonances of -₇℃ on d³RH⟺λn≥41 for all scattering resonances of −ΔH3 on Γd3 The Wheeler–DeWitt framework of canonical quantum gravity provides independent physical motivation: cosmic stability under the Hartle–Hawking no-boundary proposal forces the symmetry group Γ Γ to be an arithmetic Bianchi group, making the spectral gap condition a physical necessity. The best current partial result toward this gap is due to Blomer–Brumley (2011): λn≥14−δₙ 14 - λn≥41−δ for explicit δ>0 > 0 δ>0