Adequacy Residuals and Blind-Spot Currency

Suppose a vector is observed through two families of linear measurements: a native family that we already control and a target family that we would like to control. How much of the target response is left unexplained by the native measurements? We answer this question on finite-dimensional real or complex Hilbert spaces carrying a positive semidefinite energy. The answer is the adequacy residual, a generalized Schur complement whose quadratic form gives, for each combination of target measurements, the largest squared response at energy at most one that is invisible to the native family. We show that the residual is the smallest error left by any linear prediction of the target from the native measurements, that it vanishes exactly when the target factors through the native family, and that it obeys a Schur chain rule when measurements are added; these identities are classical facts about generalized Schur complements in the present notation. We then give explicit, checkable conditions for three practical tasks: shrinking the residual by a fixed fraction with added measurements, deciding exactly which directions an isometric restriction of the space must keep, and recovering an estimate after it has been transported to another space. An appendix treats sequences of such estimates. For a bounded native map L on a real Hilbert space and the identity as target, we take the residual to be projection onto the kernel of L; in finite dimensions this agrees with the Schur residual, and in any dimension a state v whose native part has squared norm at most K and whose residual part has squared norm at most q‖v‖² with q<1 satisfies ‖v‖² ≤ K/(1 − q). Energies may be singular, provided the measurements vanish on zero-energy vectors. Every numbered result is formalized in Lean 4 with Mathlib.