Statistical structure and physical interpretation of the fillout factor distribution in contact binary stars
Contact binary systems exhibit complex evolutionary behavior governed by mass transfer, angular momentum loss (AML), and Roche geometry. The fillout factor ($f$) is widely used as a descriptor of the degree of overcontact, yet its physical and statistical role remains nonuniformly defined across the literature. We used a compiled observational sample of W UMa-type contact binaries to construct a data-driven classification framework based on $f$ and its relation to fundamental system parameters. Unsupervised clustering applied to the one-dimensional $f$ distribution reveals a robust three-class structure corresponding to shallow-, medium-, and deep-contact configurations. The statistically determined boundaries are located at $f \simeq 0.257$ and $f \simeq 0.561$, with a stable silhouette score of $S \simeq 0.613$. The stability of this classification is confirmed through perturbation tests in $f$ and parameter-conditioned samples. A random forest regression model shows that 60.7\% of the variance in $f$ can be explained by the considered physical and geometric system parameters, with geometric parameters providing the largest predictive contribution. Among the considered parameters, the mass ratio emerges as the dominant predictor. The identified subclasses therefore appear to reflect a continuous geometric sequence rather than discrete physical states, governed by the progressive evolution of the Roche equipotential configuration under mass transfer and AML. Notably, the second boundary separating medium- and deep-contact systems shows lower stability than the first boundary, suggesting that the transition toward deep contact may be more sensitive to the physical composition of the analyzed sample.