Trinity of Varentropy: Finiteness, Fluctuations, and Stability in Power-Law Statistics

Abstract

Power-law distributions are widely observed in complex systems, yet establishing their thermodynamic consistency remains a theoretical challenge. In this paper, we present a thermodynamic framework for power-law statistics based on the \textit{renormalized entropy} $s_{2-q}$. Derived from the asymptotic scaling of the combinatorial $q$-factorial, this quantity yields a stable thermodynamic limit, remaining finite ($O(N^0)$) for systems with strong correlations. Furthermore, we clarify the physical origin of the nonlinearity parameter $q$ through the concept of \textit{Varentropy} (Variance of Entropy). By mapping the microscopic combinatorics sequentially onto the probability constraints of Type-B superstatistics, we prove that the entropic index is strictly governed by the exact identity $q-1 = 1/C$, where $C$ is the total thermodynamic heat capacity of the system and the reservoir. This deductive derivation, enforced by the uniqueness of the inverse Laplace transform, demonstrates that power-law statistics emerges as a necessary consequence of finite thermal environments, providing a first-principles description beyond the standard infinite Boltzmann-Gibbs limit ($C \to \infty$).