Local Calibration Is Not Enough: Coherent Spatial Dependence for Query-Level Uncertainty

Probabilistic models for spatial fields can appear well calibrated locally while giving unreliable uncertainty for downstream spatial queries. We study what is lost when local predictive uncertainty is preserved but cross-region dependence is discarded. For linear queries, we derive the exact factorization $R_w=N_{\mathrm{eff}}(w)\kappa (w)$, separating effective spatial participation from cross-region coherence. On Western Australian sea-surface temperature, removing cross-patch sample correspondence reduces nominal-$90\%$ domain-mean coverage from $89.5\%$ to $20.4\%$ and underestimates its variance by a factor of $46.65$. A leading coherent mode explains only $9.99\%$ of total field variance but $99.7\%$ of domain-mean variance; removing it changes $R$ from $46.65$ to $0.14$. The effect is also present in stochastic PDE reference distributions and is reproduced by two distinct generative architectures. These results show that query-level uncertainty depends critically on coherent spatial structure, motivating scalable uncertainty representations that preserve local uncertainty alongside coherent large-scale structure.