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Belief Propagation (BP) has operated in Euclidean space for four decades, yet the polynomial volume growth of $\mathbb{R}^d$ is fundamentally mismatched to hierarchical graphs: faithful embedding of exponentially branching structure demands $d = \Omega(\log n)$ dimensions and prohibitive $\mathcal{O}(d^3)$ covariance costs, while truncating $d$ corrupts marginal estimates. We introduce Continuous Hyperbolic Belief Propagation (CHBP), formulated natively on the Lorentz hyperboloid $\mathbb{H}^n_K$, whose exponential volume growth eliminates this bottleneck. CHBP parameterises beliefs as Jacobian-corrected Wrapped Normals, approximates message integrals via Gauss--Hermite quadrature, and transports covariance tensors between tangent spaces via closed-form Levi-Civita parallel transport, with a curvature annealing schedule ensuring stable convergence. On hierarchical graphs, CHBP at $d{=}5$ reduces marginal KL divergence by $25\times$ versus Euclidean BP at $d{=}50$ and achieves up to 91.45\% accuracy on real-world taxonomies, outperforming all baselines including Hyperbolic GCNs. On flat topologies, CHBP predictably underperforms Euclidean methods, confirming a topology-specific rather than universal advantage.