Hyperbolic Completion of Newton's Off-Center Orbit Problem: $SO(2,1)$ Symmetry, Inversion Duality, and Magnetic Classification

We resolve the hyperbolic off-center-orbit problem for the singular potential $$V(r)=-\frac{\alpha}{(R^2-r^2)^2},\qquad \alpha>0.$$ At zero energy, the Jacobi metric has constant negative curvature on both components separated by $r=R$. The interior Jacobi metric is a constant multiple of the Poincar\'e disk metric, while circular inversion maps the exterior isometrically to the punctured disk. We classify all zero-energy trajectories: nonradial orbits are arcs of Euclidean circles orthogonal to $r=R$, radial trajectories lie on lines through the origin, and the force center lies outside every nonradial supporting circle. An explicit Runge--Lenz-type moment map closes into $\mathfrak{so}(2,1)$, whose Casimir is the hyperbolic geodesic Hamiltonian. The canonical cotangent lift of inversion preserves the symmetry generators and maps the zero-energy flow to itself up to positive time reparametrization. The singular circle is reached in finite Newtonian time but lies at infinite Jacobi distance. Quantum mechanically, we distinguish the St\"ackel coupling transform from a genuine unitary equivalence, whose Euclidean representative is a divergence-form operator rather than the naive flat Schr\"odinger operator. The bottom of the hyperbolic continuum maps to the Hardy/oscillation threshold of the inverse-square boundary model. Finally, the symmetry-preserving radial magnetic field becomes a constant intrinsic field on the hyperbolic plane. Its shifted Casimir classifies the trajectories as closed magnetic circles, horocycles, or open hypercycles, with zero-field geodesics as the limiting case and a transition at $Q^2=8m\alpha R^2$. Numerical integrations confirm the orbit equations and conserved quantities.