Abstract
Gaussian inference on smooth manifolds is central to robotics, but exact marginalization and conditioning are generally non-Gaussian and geometry-dependent. We study tangent-linearized Gaussian inference and derive explicit non-asymptotic W_2 stability bounds for projection marginalization and surface-measure conditioning. The bounds separate local second-order geometric distortion from nonlocal tail leakage and, for Gaussian inputs, yield closed-form diagnostics from (\mu,\Sigma) and curvature/reach surrogates. Circle and planar-pushing experiments validate the predicted calibration transition near \sqrt{\|\Sigma\|_{\mathrm{op}}}/R\approx 1/6 and indicate that normal-direction uncertainty is the dominant failure mode when locality breaks. These diagnostics provide practical triggers for switching from single-chart linearization to multi-chart or sample-based manifold inference. Code and Jupyter notebooks are available at https://github.com/mikigom/StabilityTLGaussian.
