R-layer Mode Theory XXVI: Mathematical Structure of R-waves — Dynamical Foundations of the R-layer Hierarchy

This volume develops the mathematical structure of R-waves, the fundamental dynamical modes of the tension-field that propagate information across the R-layer hierarchy. Building upon the structural and cosmological foundations established in previous volumes, we introduce the first formal definition of R-waves as multi-resolution informational flows governed by deep gradients, intermediate stabilizing modes, and global curvature. We derive the general propagation equation for R-waves, incorporating fine-scale, mid-scale, and large-scale operators together with cross-scale coupling terms. This framework reveals that R-waves are not simple oscillations but structured, adaptive flows whose internal geometry determines the evolution of the tension-field. We further analyze the stability of R-waves and show how their multi-resolution decomposition enables coherent propagation across the entire hierarchy, linking quantum fluctuations, classical trajectories, and gravitational curvature. This volume establishes the dynamical foundation of the R-layer Mode Theory and prepares the ground for analytic, numerical, and physical developments in subsequent volumes.