Conditional Recovery of the Gaussian Curvature Three-Jet from Finite Geodesic Distances

We consider the recovery of the Gaussian curvature and its covariant derivatives through third order at a point of an oriented Riemannian surface. The data consist of twenty finite squared geodesic distances in a calibrated configuration: twelve right-isosceles observations at two scales and eight antipodal observations with unequal radial arms. The reconstruction separates the curvature jet into angular harmonic blocks, extracts the required distance coefficients by even–odd combinations, and corrects the nonlinear terms before inversion. Under a common strongly convex normal-neighborhood assumption, deterministic error bounds for the measured squared distances, and independently certified bounds on the relevant eighth and ninth radial derivatives, the resulting estimator satisfies an explicit weighted error bound for all ten jet coordinates. The high-derivative bounds are additional quantitative information; they are not inferred from the twenty observations. For the same fixed protocol, we also exhibit smooth complete metrics that have identical twenty squared distances but different curvature at the base point. Thus the finite measurements support conditional quantitative recovery, while exact identification from those measurements alone fails in the stated smooth class. The record includes a four-page technical supplement deriving the seventh-order distance coefficients used in the main manuscript.