This article develops a jurisprudential account of delegation to stochastic decision architectures. Existing work on automated public decision-making has already established that algorithmic systems can disrupt familiar doctrines of delegation, discretion, accountability, and administrative responsibility. The unresolved problem addressed here is narrower. A generative or otherwise stochastic system may be institutionally authorized even though, under fixed legally relevant inputs, its operative procedure can instantiate more than one legally consequential output. The legal question is therefore not exhausted by asking whether authority was delegated to an algorithm, whether a human remains nominally in the loop, or whether the system is explainable. It is whether authorization of the decision-generating architecture entails authorization of each particular output that the architecture can produce. The paper calls the discontinuity between these two levels the Distributional Authorization Gap. Let G denote an authorized generative procedure composed of a model M, a selection or decoding rule S, and institutional constraints C. For legally relevant input X, G induces a distribution P(O | X; G) over possible outputs. Institutional authorization of G does not, without a further rule of legal imputation, entail legal authorization of every oᵢ in the support of that distribution. This proposition is especially important where two technically valid executions under materially identical facts and law can produce legally non-equivalent outcomes, such as grant and denial, detention and release, eligibility and ineligibility, or sanction and non-sanction. The argument is analytical rather than empirical. It reconstructs the presuppositions of delegation theory around identifiable authority, institutional continuity, imputability, and bounded competence; distinguishes the present problem from established debates on algorithmic delegation, randomness, discretion, and predictive uncertainty; and identifies the point at which stochastic generation changes the object that must be legally authorized. The paper does not claim that randomization is inherently unlawful, that all machine-learning systems are stochastic at inference, or that legal decision-making has historically required deterministic reasoning. Law has long used lotteries, probabilistic evidence, structured discretion, and decision under uncertainty. The distinctive problem arises when an institution authorizes a generative procedure whose output space includes legally incompatible realizations without specifying why the realization actually sampled, rather than another equally procedurally available realization, counts as the institution's authorized act. The article therefore reframes stochastic delegation as a problem of output-level validity. Its central claim is that architecture-level authorization and output-level authorization are analytically distinct and that legal systems require an explicit bridge between them. That bridge may be supplied by bounded output design, a legally specified selection rule, meaningful human ratification, reviewable reasons, or another institutional rule of imputation. Where no such bridge exists, the system may be technically authorized while the legal status of its instantiated decision remains underdetermined.