An Extremal Reconstruction Principle under Covariance Domination

Abstract

We identify a structural extremal principle governing residual \(L^2\)-norms over operator-ordered covariance envelopes. In contrast to the centered setting, where such quantities reduce to trace expressions involving covariance operators, the non-centered framework generates mixed terms that cannot be recovered from covariance ordering alone. We show that the worst-case squared residual \(L^2\)-norm over an operator-ordered covariance envelope is attained at a canonical envelope representative, possibly belonging only to the closure of the admissible class. The resulting extremal identity holds uniformly over all admissible reconstruction operators. The result is obtained without convexity, compactness, or a global Hilbert space structure governing all components of the system. As a consequence, the associated minimax reconstruction problem over covariance envelopes reduces to evaluation at a canonical representative under covariance domination.