Wavelet covariances under fixed finite observations: dilation symmetry breaking and distinct endpoint limits

This preprint studies how fixed finite observations affect wavelet covariances of Gaussian fields and processes. For a homogeneous generalized Gaussian field under the stated admissibility and noise assumptions, normalized posterior wavelet covariances recover the prior covariance at both dilation endpoints, uniformly on compact target sets. Global dilation symmetry is preserved if and only if all observation functions vanish. Additional results give coefficient-level recovery rates under finite Fourier-order assumptions, a projection-norm criterion for closed Gaussian observation subspaces, and distinct posterior endpoint kernels for a finite-variance Matérn-type process with fixed noisy point observations. This record includes the manuscript PDF and a reproducibility archive containing the LaTeX source, figure-generation scripts, numerical data, verification results, and reproduction instructions.