Radial Integral Reformulation of the Gauss-Bonnet Weak Deflection Angle at Finite Distance

Abstract: We develop a radial integral reformulation of finite distance gravitational lensing in optical geometry for static, spherically symmetric spacetimes. Starting from the Gauss-Bonnet characterization of the finite distance deflection angle, we adopt the Li-type curvature primitive identity [https://doi.org/10.1103/PhysRevD.101.124058] [2006.13047], which reduces the curvature-area contribution to a one-dimensional integral evaluated along the physical light ray. We then remove the remaining implicit orbit dependence by an explicit change of variables using the null first integrals, converting the Li line integral from $φ$-integration to $r$-integration and splitting the trajectory at the turning point (closest approach). The resulting formula expresses the deflection angle as a sum of two radial integrals over $[r_0,r_S]$ and $[r_0,r_R]$ plus the finite distance angular bookkeeping term, with a transparent normalization/cancellation structure for the curvature primitive. In Schwarzschild gauge, we provide a weak-field evaluation toolkit that reduces the computation to reusable families of standard radial integrals and gives compact expressions for the endpoint incidence angles in the optical metric. Worked examples include Schwarzschild and Kottler (Schwarzschild-de Sitter) spacetimes and a black hole immersed in perfect-fluid dark matter with a finite halo. For the finite-halo model we derive closed-form leading weak-deflection expressions for mixed endpoint configurations (source/receiver inside or outside the halo), illustrating the modularity of the radial Gauss-Bonnet pipeline.