When Is a General Factor Distinguishable? Non-Proportionality, Stable Structure, and the Bifactor Decision

Whether an additional general dimension is necessary beyond correlated first-order factors is a property of the population covariance matrix, not of any estimator or design. This research establishes when that property can be decided. Where the general and group loadings are proportional within every cluster the bifactor structure is covariance-equivalent to correlated factors, so no sample size separates them (Proposition 1); where that proportionality fails in every cluster, three items per cluster and some mild regularities leave no $K$-factor model with diagonal uniquenesses able to reproduce the covariance matrix (Theorem 1); and between them lies a mixed boundary, located numerically here and turning on cluster resistance. Distinguishability is therefore graded, measured by the population distance to the $K$-factor class. Because that question is conditional on a first-order structure which is itself uncertain, a two-step procedure is developed within partially exploratory factor analysis, delivering a structure only when it reproduces across adjacent counts and treating non-delivery as legitimate. Simulation shows that a unanimous count can accompany a structure that fails to reproduce, and that absorbed local dependence can imitate a general factor, the error growing with sample size while stability indicators stay clean. Four empirical datasets illustrate the possible outcomes.