Algebraic Diversity: Group-Theoretic Spectral Estimation from Single Observations

ArXi:2604.03634v1 Announce Type: new We prove that temporal averaging over multiple observations can be replaced by algebraic group action on a single observation for second-order statistical estimation. A General Replacement Theorem establishes conditions under which a group-averaged estimator from one snapshot achieves equivalent subspace decomposition to multi-snapshot covariance estimation, and an Optimality Theorem proves that the symmetric group is universally optimal (yielding the KL transform.