When Does a Remaining Privacy Budget Preserve Every Admission Decision?

A reusable privacy budget should preserve the choice of the next release. We ask when one experiment represents exactly the requests admitted by the original spent–cap comparison after independent composition. Every cap with a nonempty feasible set has a universal residual if and only if the spend has at most one finite privacy-loss value, with complete reveal as the degenerate case. The key is how composition acts on request profiles. One value rescales the whole profile; several values average different rescalings and can conceal a crossing. We realize this obstruction with valid, bounded-likelihood-ratio experiments for every finite-support multi-atom spend, with ordered compositions and a cap equal to one of those products. Weakly continuous composition semigroups with finite privacy-loss support at each time have the one-value structure, making parameter subtraction exact for arbitrary requests. A concrete Gaussian spent–cap pair has no universal residual and admits a randomized-response channel that its Gaussian-family residual excludes. An exact finite example visualizes the obstruction, and a model-selection illustration separates the additional channel accuracy from its downstream utility. The results identify when a single remaining-budget experiment preserves the choice of the next mechanism.