A Utility-theoretic Asymptotic Equipartition Property and the Geometric Coding Theory of Risk

We develop a utility-theoretic analogue of Shannon’s asymptotic equipartition property. Utility-typical sets are defined through empirical utility averages of IID loss trajectories, and concentration is established via the strong law of large numbers and large deviation theory. We then extend the theory by introducing a geometric organization of the utility-atypical sector analogous to the structured atypical-sector program in quantum source coding. The atypical utility sector is shown to admit shell decompositions, geometric clustering, dominant atypical components, and utility-distortion coverings. This produces a geometric coding theory of risk in which preference structure induces asymptotic geometry on trajectory space and modifies the operational tradeoff between coding rate and decoding error.