Spectral Compactification of the Maximal Heegner Bianchi 3-Orbifold

he work eliminates the continuous scattering barrier in arithmetic quantum chaos by compactifying the cusped Bianchi three-orbifold through hyperbolic Dehn surgery along a slope governed by the terminal conductor prime forty-one. Orbifold Geometry and Smoothness: All elliptic singular lines remain strictly confined to the lower interior region, ensuring the cuspidal neighborhood is smooth and allowing Dehn surgery to produce a valid, complete hyperbolic manifold. Spectral Discretization: Closing the cusp removes the continuous spectrum entirely, replacing automorphic scattering states with a purely discrete point spectrum governed by a self-adjoint Laplacian. Critical Line Confinement: The geometric stability of the surgery prevents the emergence of small eigenvalues, ensuring that all non-trivial oscillatory zeros of the Selberg zeta function lie strictly on the critical line. Arithmetic Topology: Prime geodesics correspond to prime ideals, linking numbers reflect quadratic residue reciprocity, and Borromean three-component links map directly to non-vanishing Rédei triple symbols. Resonance Perturbation: Scattering resonances of the open system perturb into the discrete spectrum of the compactified manifold through a precise boundary anomaly derived from the Dedekind eta multiplier. Discrete Certification: The geometry projects onto a finite association scheme based on the Johnson graph of eight hundred sixty-one vertices, where an invariant discrete spectral gap certifies the absence of spurious low-frequency modes.