A Diagonal-Jet Criterion for Normalized Gaussian Analytic Covariances

Let \(R\) be a twice continuously differentiable, positive semidefinite complex kernel with unit diagonal on a connected plane domain. We give a necessary and sufficient condition for \(R\) to be the normalized covariance of a centered proper Gaussian analytic function with positive variance everywhere. A nonnegative residual formed from diagonal derivatives must vanish, and a real one-form determined by the first diagonal derivative must be exact. On a simply connected domain the second condition is a local closedness test. A sharp Gram inequality propagates the diagonal residual condition without division by any off-diagonal kernel value. The covariance is reconstructed up to a positive constant. A polynomial example has a zero in every base section, and an annular example separates local conditions from the global period obstruction. This elementary note concerns the full complex normalized covariance, not arbitrary zero-process correlations. It uses classical positive-kernel and Gaussian-series methods and makes no absolute priority claim.