Spinoptical weak deflection from Gauss–Bonnet lensing with reference and ribbon geometries

We develop a finite-distance weak-lensing framework for helicity-dependent light propagation (spinoptics) in static spherically symmetric spacetimes. Because spinoptics generically drives rays away from a fixed plane, the natural local observable is a two-component change of direction on an endpoint celestial sphere, while the Gauss–Bonnet construction computes an endpoint-subtracted finite-distance in-plane component. We combine this endpoint observable with a nonplanar ribbon domain ruled by radial geodesics. The ribbon construction preserves the curvature-area reduction of optical Gauss–Bonnet lensing while retaining the boundary contribution generated by non-geodesic propagation. To avoid orbit-based normalization, we fix the additive gauge of the curvature primitive by reference subtraction, using Minkowski space for asymptotically flat cases and the de Sitter static patch for Kottler. This yields photon-sphere-free expressions for the in-plane finite-distance component as a one-dimensional primitive integral. The helicity-dependent sector is expressed through line integrals, but two out-of-plane quantities must be distinguished: [Formula: see text] is a one-sided post-pericenter measure of accumulated normal drift, whereas [Formula: see text] is the genuine source-to-receiver endpoint observable. We implement the formalism explicitly in a leading-order Schwarzschild case and then present Reissner–Nordström, Kottler, and Janis–Newman–Winicour as illustrative weak-field extensions showing how charge, a cosmological constant, and scalar hair can enter the same decomposition. Within this extension-level treatment, the Kottler case isolates mixed helicity–mass–cosmological-constant terms arising from the de Sitter reference-orbit geometry rather than from direct local [Formula: see text]-dependent spin–curvature coupling.