Compositional Structural Metrics on Atomic Transition Forms

This work introduces the class of compositional structural metrics (CSMs), a family of quantitative invariants defined on the factor space of atomic forms and compatible with both sequential composition and structural embedding. The paper formalizes the axioms of isomorphism invariance, embedding monotonicity, subadditivity, and representation independence, and shows that atomic length and atomic depth are fundamental examples of such metrics. It further analyzes derived structural quantities, structural trade-offs between different invariants, closure properties of the CSM class under nonnegative linear combinations, and the pseudometric structures induced by CSMs. The results provide a quantitative layer over the structural algebra of atomic dependency forms and prepare the ground for geometric and metric analysis of compositional computational structures.