R-layer Mode Theory XXVI-B: Unified Hierarchical Geometry and the Full R-wave Equation

R-layer Mode Theory Volume XXVI-B completes the construction of the R‑wave theory by integrating the quantum, informational, and gravitational layers into a single unified hierarchical geometry. Building on the analytical foundations of Volume XXVI‑A, this volume introduces the unified operator, the hierarchical geometry GR, and the full non-linear, multi-scale R-wave equation. As stated in the introduction: “The central objective of this volume is to construct the unified operator Dunified that governs the full R-wave dynamics across all layers.” The volume develops the geometric structure of each layer: Quantum geometry (deep layer): quantum metric, quantum curvature, and geometric form of the deep operator Informational geometry (mid layer): informational metric, informational connection, informational curvature, and bidirectional coupling Gravitational geometry (global layer): gravitational metric, cosmological curvature, and curvature-driven R-wave propagation These components are integrated into: the unified operator Dunified, combining three geometric operators and four bidirectional coupling operators the hierarchical geometry GR=(gQ,gI,gG)+CR, defining metrics, connections, and unified curvature the full R-wave equation Dunified Y=0, the complete dynamical expression of the theory The volume also develops: hierarchical curvature flow w-layer geometric circulation multi-metric numerical integration schemes unified numerical stability conditions observational predictions (filament spacing, satellite well periodicity, curvature imprints, large-scale coherence) Volume XXVI‑B provides the complete geometric and dynamical formulation of the R-wave theory and prepares the observational analysis developed in Volume XXVI‑C.