Fixed-order perturbative calculations for differential cross sections can suffer from non-physical artifacts: they can be non-positive, non-normalizable, and non-finite, none of which occur in experimental measurements. We propose a framework, the Resummed Distribution Function (RDF), that, given a perturbative calculation for an observable to some finite order in αs, will « resum » the expression in a way that is guaranteed to match the original expression order-by-order and be positive, normalized, and finite. Moreover, our ansatz parameterizes all possible finite, positive, and normalized completions consistent with the original fixed-order expression, which can include NnLL resummed expressions. The RDF also enables a more direct notion of perturbative uncertainties, as we can directly vary higher-order parameters and treat them as nuisance parameters. We demonstrate the power of the RDF ansatz by matching to thrust to O(α3) and extracting αs with perturbative uncertainties by fitting the RDF to ALEPH data.