On the uniqueness of the coupled entropy

The coupled entropy, $H_κ,$ is proven to uniquely satisfy the requirement that a generalized entropy be a measure of the uncertainty at the scale, $σ,$ for a class of non-exponential distributions. The coupled stretched exponential distributions, including the generalized Pareto and Student's t distributions, are uniquely parameterized to quantify linear uncertainty with the scale and nonlinear uncertainty with the tail shape for a broad class of complex systems. Thereby, the coupled entropy optimizes the representation of the uncertainty due to linear sources. Lemmas for the composability and extensivity of the coupled entropy are proven. The uniqueness of the coupled entropy is further supported by demonstrating consistent thermodynamic relationships, which correspond to a model used for the momentum of high-energy particle collisions. Applications of the coupled entropy in measuring statistical complexity, training variational inference algorithms, and designing communication channels are reviewed.