Mathematical Theory of Organization and Structure: Foundations, Transformations, and the Map of the Theory

This book presents the foundations of the Mathematical Theory of Organization and Structure (TMOE), a mathematical framework for studying how organizational structure arises, changes, and can be compared under restrictions, transformations, refinements, and changes of description. The theory is developed from a minimal relational organization O = (Ω, ↝), where Ω denotes a space of admissible configurations and ↝ denotes an elementary transition relation. From this starting point, the book develops accessibility, coaccessibility, organizational regimes, condensation, refinement, organizational morphisms, isomorphism, projection, observables, compression, reconstruction, iterated evolution, and asymptotic behavior. The later chapters formulate a principle of minimal extension and examine connections with dynamical systems, discrete structures, spectral and advanced dynamics, and the Residual Structural Core (RSC). These connections are treated as extensions or applications of the foundational framework rather than as replacements for the theories involved. The book also identifies the limits of the present framework, formulates open mathematical problems, and defines a program for its further development through mathematical investigations and specialized research articles. Book I is intended as the foundational and reference volume of the TMOE research program.