R-layer Mode Theory Appendix Y: Mathematical Structures of R-layer Mode Theory

Appendix Y develops the geometric foundations of the R-layer Mode Theory (RLMT), providing the structural framework that underlies the layered decomposition of fields and operators. Whereas Appendix X establishes the algebraic specification of AUP, MUP, and CUP modes, Appendix Y focuses on the geometric interpretation of layered structures, the fiber-like organization of mode spaces, and the correspondence rules that relate geometric data across layers. The appendix introduces the geometric primitives required for RLMT, including layered manifolds, inter-layer maps, and the structural relations that govern how geometric information propagates through the R-layer hierarchy. The document formalizes the geometric constraints that ensure consistency between algebraic mode definitions and their spatial or configuration-space realizations. It clarifies how layered geometry supports the construction of effective fields, projection structures, and geometric correspondences that appear throughout the RLMT series. By establishing a unified geometric language, Appendix Y provides the conceptual bridge between the abstract operator-level rules of Appendix X and the physically interpretable constructions used in higher RLMT volumes. Designed to be self-contained, Appendix Y serves as the geometric baseline for all subsequent developments in RLMT, including cosmological applications, quantum-informational structures, and geometric-mode correspondences. Together with Appendix X, it completes the foundational layer of the theory, ensuring that all RLMT volumes share a consistent mathematical and geometric infrastructure.