Strictly Proper, Yet Mutually Singular: Cameron--Martin Barriers and Minimax Score Design for SPDE Law Operators

Strict propriety identifies the data-generating law at zero population regret, but not the topology induced by finite-sample near-minimization. For a parabolic SPDE law operator $K_\theta(x)=\Normal(M_\theta x,C)$, we derive an exactly equivalent Gaussian-sequence experiment and incompatible geometries: Wasserstein and energy losses weight errors by the trace-class covariance spectrum, whereas Hellinger loss uses the unweighted Cameron--Martin norm. A bounded estimator therefore attains $O(n^{-1})$ Wasserstein and energy risks while its forecast is mutually singular with the truth almost surely for every finite $n$. Over a Sobolev ellipsoid, the architecture-independent minimax rates are $n^{-1}$ for the weak losses and $n^{-2\beta/(2\beta+1)}$ for Hellinger loss. We prove the sharp calibration modulus $\omega_{E\to H}(\epsilon)\asymp R^{2\alpha/(\alpha+\beta)}\epsilon^{\beta/(\alpha+\beta)}$ and a compactness obstruction: every twice-regular score based on a sample-finite Gaussian-linear observation has trace-class Fisher-normalized curvature. The resulting finite-information problem has an exact water-filling solution, and matched score--regularization procedures attain the Hellinger minimax rate. For fractional parabolic SPDEs on a $d_{\mathrm{sp}}$-dimensional domain, this rate becomes $n^{-2s/(2s+d_{\mathrm{sp}})}$.