Suppose we wish to determine the quality of a candidate solution to a convex stochastic program in which the objective function is a statistical functional parameterized by the decision variable and known deterministic constraints may be present. Inspired by stopping criteria in primal-dual and interior-point methods, we develop cancellation theorems that characterize the convergence of appropriately resampled and standardized primal and dual objective values to a weak limit. The resampled weak limit is distribution free, meaning it does not depend on the data-generating distribution. Furthermore, it is expressed as a functional of the standard Brownian motion, facilitating its implementation and valid use in optimality gap confidence interval construction and KKT point statistical testing. Since our results are general, we anticipate their use in iterative algorithm termination criteria for stochastic linear programs, two-stage convex stochastic programs, and a variety of finite- and infinite-dimensional problems arising in maximum likelihood estimation, nonlinear regression, classification, and portfolio management.