When Does Learned Posterior Geometry Recover Weighted Manifold Forms?

When does accurate local posterior geometry yield accurate weighted function energies? We study a fixed ambient trial space, separating learning error, finite-noise geometric bias, and numerical integration. A gradient-dependent bound connects local matrix error to the weighted form, while circle examples, including an exact Gaussian observation posterior, show how small normal components produce large form errors through ambient extensions. On controlled image-translation manifolds, we compare matched covariance models trained with sample residuals or conditional second moments. These objectives have the same ideal population minimizers but different conditional label randomness. With oracle-assisted centering and known-dimensional filtering, conditional supervision reduces mean linear-form error from $24.80\%$ to $3.73\%$, with improvements numerically distinguished in all four image–initialization pairs. Same-form attribution identifies learning-induced normal error, rather than finite-noise posterior variance, as the dominant raw discrepancy in this cohort. Improving a learned mean instead improves tangent directions, worsens restricted metric scales, and produces large opposing fitting and centering contributions. Together, these findings connect extension-sensitive posterior forms, same-objective supervision effects, and signed geometric errors. They help researchers choose supervision and ambient probes by their effect on weighted energies, beyond aggregate local matrix accuracy.