The coupled entropy is proven to uniquely satisfy the requirement that a generalized entropy be equivalent to the density at the scale for scale-shape distributions. Further, its maximizing distributions, the coupled stretched exponential distributions, are proven to quantify the linear uncertainty with the scale and the nonlinear uncertainty with the shape for a broad class of complex systems. Distributions of the coupled exponentials include the Pareto Types I-IV and Gosset's Student-t. For the Pareto Type II distribution, the Boltzmann-Gibbs-Shannon entropy has a linear dependence on the shape, which dominates over the logarithmic dependence on the scale, motivating the need for a generalization. The Rényi and Tsallis entropies are shown to be of historic importance but ultimately unsatisfactory generalizations. The coupled entropy of the coupled stretched exponential distribution isolates the nonlinear-shape dependence to a generalized logarithm of the partition function. The Rényi and Tsallis entropies retain a strong dependence on the nonlinear-shape such that they are not equivalent to the uncertainty at the scale. Lemmas for the composability and extensivity of the coupled entropy are proven in support of an axiomatic definition. The scope of the coupled entropy includes systems in which the growth of states is power-law, stretched exponential, or a combination.

翻译：耦合熵被证明唯一满足广义熵在尺度-形状分布中与尺度密度等价的要求。进一步地，其最大化分布——耦合拉伸指数分布——被证明能够量化广泛复杂系统中尺度的线性不确定性与形状的非线性不确定性。耦合指数分布包括帕累托I-IV型分布和高斯学生t分布。对于帕累托II型分布，玻尔兹曼-吉布斯-香农熵对形状具有线性依赖性，该依赖性主导了对尺度的对数依赖性，这推动了对广义熵的需求。研究表明，Rényi熵和Tsallis熵具有历史重要性，但最终作为广义化方案并不令人满意。耦合拉伸指数分布的耦合熵将非线性形状依赖性隔离至配分函数的广义对数中。Rényi熵和Tsallis熵仍保持对非线性形状的强依赖性，导致其无法等价于尺度不确定性。本文证明了耦合熵的可组合性与广延性引理，以支持其公理化定义。耦合熵的适用范围包括状态增长呈幂律、拉伸指数或两者组合的系统。