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

For a centered homogeneous generalized Gaussian field with spectral weight [Formula: see text], we prove two-endpoint recovery of normalized posterior wavelet covariances under fixed finite admissible observations. Global dilation symmetry holds if and only if all observation functions vanish. The nonzero wavelet and the observation functions are real Schwartz functions with [Formula: see text] vanishing moments, with [Formula: see text] and [Formula: see text]. Noise is Gaussian and field-independent, and total observation covariance is positive definite. Under simultaneous dilation of target scales and positions, multiplication by [Formula: see text] recovers the prior covariance as [Formula: see text] and [Formula: see text], uniformly on every fixed compact target set. If a nonzero observation function exists, additional finite first nonzero Fourier orders [Formula: see text] at zero of the wavelet and nonzero observation functions yield compact-uniform recovery bounds [Formula: see text] and [Formula: see text], with leading coefficients and cancellation-dependent improvements. The bounds are exact leading orders when their coefficients are nonzero. For the Matérn-type spectrum [Formula: see text], [Formula: see text], [Formula: see text], fixed finite noisy point observations yield distinct compact-uniform limits: the homogeneous [Formula: see text] kernel after multiplication by [Formula: see text] at small scales, and the constant-spectrum [Formula: see text] kernel without normalization at large scales. A projection-norm criterion treats fixed closed observation subspaces; Mexican-hat formulas illustrate the finite-observation results.