R-layer Mode Theory (RLMT) XXVI-A: Analytical Development of the R-wave Theory

R-layer Mode Theory Volume XXVI-A develops the analytical foundations of the R‑wave theory, a hierarchical excitation that propagates across the deep (quantum), mid (informational), and global (gravitational) layers. This volume establishes the three-layer operator system, derives the analytical R-wave equation, and clarifies the stability conditions required for hierarchical propagation. The analysis begins with the spectral structure of the three operators Ddeep,Dmid,Dglobal, demonstrating strict hierarchical ordering of eigenvalues. As stated in the volume: “The analytical task of this volume is to show how the three-layer operator system emerges, how its spectral structure is organized, and how the R-wave equation arises as a natural consequence of hierarchical coupling.” The volume then derives the unified analytical operator Danalytic, showing how deep-layer fluctuations seed the mid layer, how informational curvature mediates between scales, and how global modes generate coherent cosmological structure. Key analytical results include: hierarchical spectral ordering and degeneracy derivation of the analytical R-wave equation unified stability conditions across all layers analytical formation of filaments, voids, and satellite wells introduction of informational flow, curvature, and second-order curvature construction of the analytical-to-numerical bridge for Volume XXVI‑B This volume forms the analytical backbone of the R-wave theory and prepares the unified geometric formulation developed in Volume XXVI-B.