A unique, universal definition for the entropy of nonlinear, complex systems is proposed. The Shannon entropy is shown to measure energetic, configurational, and nonlinear degrees of freedom (DOF). An entropy calibrated to the asymptotic nonlinearity of the distribution reduces the nonlinear DOF to zero and modifies the configurational DOF with a generalized logarithm. Physical axioms regarding the asymptotic growth of states, the calibration from an informational scale, and nonlinear dynamics provide a foundation for the proof that the calibrated entropy is unique and comprehensive across the short- and long-range dependency classes. The informational scale quantifies the linear uncertainty independent of the nonlinearity. This entropy, when constrained by the informational scale, is maximized by the coupled stretched exponential distributions (CSED). A consistent thermodynamics for nonequilibrium systems is proposed with the temperature equal to the scale constraint divided by the Boltzmann constant. Physical and informational applications are highlighted.

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