In statistical mechanics, evaluating finite-size macroscopic fluctuations typically relies on Edgeworth expansions. However, these perturbative methods append additive polynomial corrections that inevitably break down in the large deviation regime, yielding unphysical negative probabilities. We propose a structural resolution: rather than relying on additive polynomials, we absorb finite-size skewness using a globally stable $q$-algebraic framework. By introducing a dynamic scaling law $1-q_n = O(n^{-1})$ for the nonextensivity parameter, we prove this $q$-deformed framework exactly captures macroscopic higher-order fluctuations in independent and identically distributed (i.i.d.) systems. Specifically, our exact algebraic tuning completely absorbs third-order skewness while structurally guaranteeing probability density non-negativity across the entire domain. Furthermore, the $k$-th degree term of this $q$-logarithmic expansion universally corresponds to the $O(n^{1-k/2})$ asymptotic order of classical $(k+1)$-th moment Edgeworth corrections. This exact correspondence functions as a stable resummation of divergent asymptotic expansions, establishing a fundamental mathematical bridge between finite-size i.i.d. fluctuations and the Tsallis statistics governing complex systems.

翻译：在统计力学中，评估有限大小的宏观涨落通常依赖于埃奇沃思展开。然而，这些微扰方法附加了加性多项式修正，在 大偏差 区域必然失效，从而产生物理上不合理的负概率。我们提出一种结构性的解决方案：不依赖于加性多项式，而是使用全局稳定的 $q$-代数框架来吸收有限大小的偏度。通过引入非广延参数的动态标度律 $1-q_n = O(n^{-1})$，我们证明这个 $q$-形变框架能够精确捕获独立同分布(i.i.d.)系统中宏观的高阶涨落。具体而言，我们的精确代数调谐完全吸收了三阶偏度，同时从结构上保证了整个域上概率密度函数的非负性。此外，这个 $q$-对数展开的 $k$ 阶项经典地与对应 $(k+1)$ 阶矩埃奇沃思修正的 $O(n^{1-k/2})$ 渐近阶数普遍对应。这种精确对应充当了发散渐近展开的稳定重求和机制，在有限大小独立同分布涨落与支配复杂系统的Tsallis统计之间建立了一座基本的数学桥梁。