The non-backtracking (Hashimoto) operator B on directed edges (i→j) captures message-passing and avoids spurious loops. The epidemic threshold is λ₁(B)⁻¹ where λ₁ is B's largest real eigenvalue. For locally tree-like networks, β/μ > 1/λ₁(B) triggers a macroscopic outbreak. This gives sharper predictions than the adjacency matrix threshold (mean-field) on heterogeneous networks.