Fisher Completions and Statistical Regularization of Wrapped Gaussian Families

We determine the intrinsic Fisher completion of the full-covariance wrapped Gaussian family on a fixed torus and relate its geometry to uniform estimation. The completion is homeomorphic to the total-variation closure. Every added law is, in suitable integer torus coordinates, a product of a uniform law and a lower-dimensional wrapped Gaussian. All these densities are positive and real analytic, but in dimension at least two no added point has a compact neighborhood. Small balls remain contractible, and explicit minimizing geodesics pass through added points. In dimension at least two, every fixed Fisher neighborhood of an added point has a positive minimax risk at every sample size. For subfamilies with a common positive lower bound on the variance in every nonzero integer direction, we prove that compactness of the completed closure is equivalent to uniform consistency in either $L^1$ or completed-Fisher loss. Equivalently, the Fourier moments must decay uniformly over the subfamily. A fixed covariance floor guarantees this condition without an upper covariance bound and gives $L^1$ risk at most $C_{\varepsilon,d}n^{-1/2}\{\log(n+1)\}^{d/4}$. A common-phase example illustrates how partially uniform limits can retain relative-phase information while remaining uniformly estimable.