A universal structure of the Skewness–Kurtosis plane for n = 4

Skewness and kurtosis are fundamental descriptors of distributional shape, widely used to characterize asymmetry and extreme fluctuations across a broad range of natural and complex systems. Their representation in the skewness–kurtosis plane provides a geometric description of distributional shape, yet the admissible domain of these quantities for finite samples remains only partially understood. In particular, for minimal sample size, numerical evidence has revealed the emergence of a deltoid-shaped region with intricate internal organization, whose exact nature has remained unexplained. Here, we show that for samples of size 𝑛 = 4, the admissible skewness–kurtosis domain is exactly described by a universal deltoid curve, whose boundary can be derived analytically and independently of the underlying distribution. Beyond the boundary, we demonstrate that the internal structure of the domain exhibits a nontrivial organization that depends on the effective sampling of the configuration space and is naturally induced by discrete representations. In particular, we show that similar structured patterns may also arise from purely random signals when discretization or finite-resolution effects are present, while the intrinsic combinatorial structure provides a minimal mechanism for their emergence. These findings reveal a previously unrecognized geometric and combinatorial structure underlying higher-order statistical moments, supported by both synthetic and real-world data. While the presence of internal patterns is not, by itself, a unique signature of a specific physical mechanism, our results provide a rigorous foundation for the interpretation of skewness–kurtosis relations in finite samples and uncover a universal organizing principle governing the shape of moment space.