Extended Tetrad Analysis in Factor Modelling: Separability and Uncertainty from Multidimensional Dependence Structures

Geometric representations provide a principled framework for structuring the description of latent constructs and clarifying sources of uncertainty in their dimensional characterisation. We introduce a novel geometric representation of factor models via two subspaces spanned by paired matrices, where determinantal expressions explicitly quantify the contributions of different dimension subsets to the factor structure. This formulation refines rank-based conditions relevant to understanding factor score indeterminacy and the implications of non-uniqueness in instrumental variable estimation for over-identified models. By weighting these multidimensional contributions to encode sensitivity to their variation, we extend the definition of tetrads into an algebraic procedure that establishes conditions for identifying variability components attributable to individual dimensions. Focusing on cases where one factor encodes structural information, we derive minimal conditions-expressed through graph planarity-that ensure such dimension-specific identifiability. The proofs yield both formal verification tools and constructive methods for generating counterexamples where these conditions fail. These counterexamples reveal a type of ambiguity, termed contextuality, in which the comparison of dimensional contributions depends on the choice of remaining reference dimensions, violating well-established axioms of order-theoretic consistency. We relate these findings to specific forms of uncertainty examined in the psychometric literature.