Closed forms and open obstructions: the sample variance of three observations
The exact distribution of the sample variance for $n=3$ is known since Rietz (1931) for the uniform parent, and Royen (2007, 2008) gave a Fourier series for any bounded continuous parent. Neither settles which parents admit a finite closed form, nor which special functions it forces. In coordinates aligned with the cube diagonal the variance constraint becomes a cylinder and the cube cross-section a polygon with $S_3$ symmetry, hexagonal over the central band of the diagonal and triangular near either corner; the CDF is the volume of their intersection, the parent density entering as a weight. A closure hierarchy is then organized by the minimal function class containing the parent density: polynomial parents on any bounded interval always close in elementary terms, a theorem for the whole class; in the negative direction a single explicit parent already suffices, and for a rational one we prove the CDF is not elementary, the obstruction being an irreducible dilogarithmic part. Beyond those two theorems the hierarchy is a set of example-specific obstructions rather than a classification: for an algebraic parent the radial first-kind differential is shown non-elementary on a genus-two curve, and the exponential row is a conjecture supported by the Bessel structure of its radial integral. For the uniform parent we obtain a two-piece formula bifurcating at $Y=1/4$, where the variance disk first circumscribes the hexagonal cross-section at the cube center. For the singular arcsine parent we derive both endpoint laws in closed form and give a six-term approximation accurate to about $10^{-3}$, an accuracy Royen's universal series reaches at about a hundred terms.