The Second Curvature Correction in the Pythagorean Residual of a Geodesic Right Isosceles Triangle

We study the Pythagorean residual of a geodesic right isosceles triangle as a scalar observable of local geometry. Let the two equal geodesic legs have length s, let c(s) denote the opposite geodesic side, and define R(s)=c(s)²-2s². For two-dimensional constant-sectional-curvature models, we derive R(s)=-(K/3)s⁴-(2K²/45)s⁶+O(s⁸). The leading coefficient agrees with the classical curvature correction for orthogonal geodesic triangles. Our contribution is the explicit order-six coefficient for this configuration and the scale-free identity C₂=-(2/5)C₁². The first coefficient determines the sign and magnitude of K, whereas the second determines its magnitude and supplies an internal compatibility check. We validate both coefficients on spherical, Euclidean, and hyperbolic models using high-precision evaluations of the exact cosine laws. The experiments also expose an intermediate extraction window: truncation error dominates at large scales, while floating-point cancellation dominates at very small scales. The claims are restricted to right isosceles triangles in two-dimensional constant-curvature spaces; variable curvature, noisy observations, and general triangle shapes are left for future work.