A mathematical theory of complex specified information is introduced which unifies several prior methods of computing specified complexity. Similar to how the exponential family of probability distributions have dissimilar surface forms yet share a common underlying mathematical identity, we define a model that allows us to cast Dembski’s semiotic specified complexity, Ewert et al.’s algorithmic specified complexity, Hazen et al.’s functional information, and Behe’s irreducible complexity into a common mathematical form. Adding additional constraints, we introduce canonical specified complexity models, for which one-sided conservation bounds are given, showing that large specified complexity values are unlikely under any given continuous or discrete distribution and that canonical models can be used to form statistical hypothesis tests, by bounding tail probabilities for arbitrary distributions.