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 $(μ,Σ)$ and curvature/reach surrogates. Circle and planar-pushing experiments validate the predicted calibration transition near $\sqrt{\|Σ\|_{\mathrm{op}}}/R\approx 1/6$ 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.

翻译：光滑流形上的高斯推断是机器人学的核心问题，但精确的边缘化与条件化通常是非高斯的且依赖于几何结构。本文研究切线性化高斯推断，并针对投影边缘化与曲面测度条件化推导出显式的非渐近$W_2$稳定性界。该界限将局部二阶几何畸变与非局部尾部泄漏分离开来，并对高斯输入情形，从$(μ,Σ)$及曲率/可达距离代理量中得到闭式诊断指标。圆环与平面推动实验验证了在$\sqrt{\|Σ\|_{\mathrm{op}}}/R\approx 1/6$附近发生的标定转变，并表明当局部性条件破坏时，法向不确定性是主要的失效模式。这些诊断指标为从单图线性化切换到多图或基于采样的流形推断提供了实用的触发判据。