This paper studies when, under restrictions on treatment assignment, departures from unconfoundedness do not invalidate empirical analyses based on that assumption. We allow treatment assignment to remain related to potential outcomes after conditioning on observed covariates, but restrict this dependence using conditional mutual information. Specifically, we require that the information about treatment assignment contained in a potential outcome, conditional on observed covariates, does not exceed the information contained in latent outcome heterogeneity. We show that this restriction is equivalent to a multiplicative factorization of the treatment probability into separate functions of latent heterogeneity and observed covariates. This characterization reveals that sufficiently rich variation in treatment probabilities with observed covariates is enough to recover unconfoundedness. For binary treatments, any nonconstant variation in the propensity score is sufficient; for multiple treatments, the corresponding requirement is a full-rank condition on the vector of conditional treatment probabilities. Treatment-state-specific versions of the restriction can recover exogeneity for individual potential outcomes under boundary- support conditions. For continuous treatments, completeness of the conditional treatment distribution plays the role of treatment probability variation and can be weakened to pointwise relevance when interest is restricted to conditional mean identification. We also extend the framework to instrumental variable models, where it provides information-theoretic relaxations of instrument exogeneity while maintaining the conventional exclusion restriction.