Deciding Response-Update Feasibility from Incomplete Covariance Information

Updating a model under explicit response budgets requires deciding whether a requested change can be fitted without excessive interference or update size. A compact covariance sketch makes this decision depend on retained information as well as on the request. We develop a three-way feasibility interface for fixed features at one linear projection. A realizable Frequent Directions construction gives identical sketches, shrink values and residual traces with opposite answers to the same request. Given additional covariance-vector actions, classical positive semidefinite (PSD) interpolation of the residual and its complement yields attainable cost endpoints for a radius-and-column completion set, uniformly over adaptively selected updates. An upper-feasible candidate and a full-residual-corrected lower value then give a witness, an exclusion or an unresolved interval for unregularized target error. The same budgets apply jointly to shared responses and to complete cumulative updates. A captured GPT-2 shared-response problem exhibits all three outcomes at one target tolerance; its intermediate request remains unresolved despite a feasible full-information reference. A saved-state analysis of 32 factual requests finds nearly equal first-witness costs for optimistic queries and strong Krylov after the initial sketch pass is counted, despite differences in fitting accuracy. Native factual evaluation and cumulative candidate rejections expose the limits of finite-feature protection. The results connect covariance information to actionable response decisions without turning candidate failure into a claim of infeasibility.