R-layer Mode Theory Appendix X: Unified AUP/MUP/CUP Mode Specification

Appendix X provides the canonical and unified specification of the AUP, MUP, and CUP mode structures that form the mathematical backbone of the R-layer Mode Theory (RLMT). This document consolidates all definitions, conventions, and operator-level rules that appear across the RLMT series, establishing a single authoritative reference for the construction of layered fields, mode decompositions, and inter-layer correspondences. The appendix formalizes the algebraic relations among the three fundamental mode types, clarifies their transformation properties, and presents the unified notation system used throughout the theory. It also defines the projection operators, normalization rules, and layer-indexing conventions that ensure internal consistency across all RLMT volumes. Beyond serving as a reference, Appendix X functions as the foundational “specification document” for RLMT. It provides the minimal mathematical structure required to reproduce any RLMT calculation, including layered propagators, effective actions, and geometric correspondences. The appendix is designed to be self-contained and independent of any particular physical model, allowing it to serve as a stable baseline for future theoretical developments. Together with Appendix Y (geometric foundations), Appendix X completes the formal infrastructure on which the higher RLMT volumes—cosmological, quantum-informational, and geometric—are built.