The Chemical Simplex: Geometric Foundations of Mixture Observability, Intrinsic Dimensionality, and Rank Constraints in Chemometrics

Chemical analysis of multicomponent systems inherently relies on the assumption that measured data correspond to chemically admissible compositions arising from a finite set of independent species.This work formalizes this assumption by deriving the geometric structure imposed by three minimal, chemically grounded constraints: (i) conservation of matter with respect to the observable (operational closure), (ii) non-negativity and finite chemical independence of species amounts, and (iii) linear additive observability.Under these conditions, the set of all physically admissible composition states forms a convex simplex of dimension 𝑘 -1, where 𝑘 denotes the number of chemically independent species.This affine structure is preserved under any linear additive mapping to observable space, regardless of the chemical nature of the system, the type of observable, or the specific data analysis method employed.As a direct consequence, the intrinsic dimensionality of any chemically admissible observable dataset is strictly bounded by 𝑘 -1, independent of the number of measurements, experimental conditions, or instrumental details.Deviations from convex simplex geometry or violations of this dimensional bound serve as objective, chemically meaningful diagnostics of violated assumptions, such as material loss or gain, unaccounted phases or species, nonlinear detection effects, or non-stationary behavior.The proposed framework constitutes a unifying geometric foundation for mixture observability, intrinsic dimensionality assessment, rank interpretation, and chemical consistency evaluation in chemometric modeling of multicomponent systems.