This paper proposes a comprehensive and unprecedented framework that streamlines the derivation of exact, compact -- yet tractable -- solutions for the probability density function (PDF) and cumulative distribution function (CDF) of the sum of a broad spectrum of mixed independent positive random variables (RVs). To showcase the framework's potential and extensive applicability, we tackle the enduring challenge of obtaining these statistics for the sum of fading variates in an exact, manageable, and unified manner. Specifically, we derive novel, tractable expressions for the PDF and CDF of the sum of Gaussian-class and non-Gaussian-class fading distributions, thereby covering a plethora of conventional, generalized, and recently introduced fading models. The proposed framework accommodates independent and identically distributed (i.i.d.) sums, independent but not necessarily identically distributed (i.n.i.d.) sums, and mixed-type sums. Moreover, we introduce the strikingly novel $α$-$μ$ mixture distribution that unifies all Gaussian-class fading models.