Recovering the Gaussian Curvature Jet from Pythagorean Residuals of Rotated Geodesic Right Isosceles Triangles

This work derives the small-scale Pythagorean residual through order six for rotated geodesic right isosceles triangles on a smooth Riemannian surface. The expansion determines the Gaussian curvature, its covariant gradient, and its covariant Hessian through finite angular-harmonic reconstruction systems. Independent coefficient-separating families establish the universal constants, and a variable-curvature numerical example validates the reconstruction.