Canonical Decompositions and Structural Invariants of Finite Architectures

Canonical decompositions are among the most powerful organizing principles in mathematics. Prime factorization, Jordan canonical form, spectral decomposition, and Hodge decomposition demonstrate that complicated mathematical objects often admit unique descriptions in terms of elementary structural components. Surprisingly, despite the ubiquity of finite relational systems throughout mathematics, no general decomposition theory appears to exist for finite architectures themselves. Existing approaches typically begin with graphs, operators, Green functions, networks, or application-specific models rather than with architecture as the primitive mathematical object.In this paper we develop a decomposition framework for finite architectures generated by four primitive structural processes: transport, interaction, closure, and hierarchy. Within the free architectural algebra introduced here, every architecture possesses a unique canonical decomposition into these primitive generators. This decomposition induces canonical architectural coordinates and leads to a representation theorem showing that every positive multiplicative structural invariant is uniquely determined by its values on the primitive generators.The framework is formulated independently of any particular realization. Graph theory, Green-function methods, operator theory, and Green-Function Architecture Theory arise naturally as representations of the same underlying structural decomposition. The resulting theory separates structural decomposition from mathematical representation and provides a foundation for studying architectural invariants independently of their realization. The extension of the decomposition theory beyond the free architectural algebra and the determination of distinguished architectural basis constants are formulated as central problems for future research.