Gauss–Bonnet lensing and boundary wave response in anti-de Sitter spacetime

We connect finite-distance Gauss–Bonnet lensing to a boundary wave response in asymptotically anti-de Sitter spacetime. Reference subtraction retains local incidence angles and displacement of the optical-curvature domain. The pure-AdS reference becomes degenerate at conformal infinity, making fixed-endpoint matching nonuniform even for weak deflection. Comparison at fixed conserved impact parameter instead yields a regular scattering characteristic on weak returning branches. For Reissner–Nordström–AdS, we reconstruct this characteristic from the curvature angle and test it against direct ingoing solutions of the conformally coupled scalar equation. An exact boundary relation fixes the first-return amplitude without a fitted normalization. Independent optical Jacobi equations recover both angular focusing directions. Gaussian-filtered boundary profiles from the wave equation agree with optical-phase harmonic sums, with relative complex errors decreasing from 8.5 × 10 −5 to 2.2 × 10 −5 over a fourfold frequency increase in the reported example. We distinguish this geometric-optics convergence from weak-field phase truncation, whose error grows with frequency.