R-layer Mode Theory XXVII: Analytical Development of R-waves Spectral Structure and Stability Across the R-layer Hierarchy

R-layer Mode Theory Volume XXVII develops the analytical foundations of R-wave dynamics, extending the operator-based framework introduced in Volume XXVI. This volume constructs the full mathematical structure of the R-wave operator, derives the hierarchical eigenmode decomposition, analyzes the coupled spectral branches, and establishes rigorous stability conditions across the R-layer hierarchy. As stated in the introduction: “R-waves arise as perturbations of the tension-field… the operator is inherently hierarchical and non-uniform: it couples multiple resolutions, admits nontrivial spectral clusters, and supports hybrid modes that propagate across scales.” The volume introduces: resolution operators Ddeep,Dmid,Dglobal cross-scale coupling operator C, generally non-self-adjoint hierarchical eigenmode decomposition with left/right eigenfunctions and biorthogonality spectral branches (deep, mid, global, hybrid) and their dispersion relations perturbative corrections and mixing-angle formulation for hybridization pseudospectral structure arising from operator non-normality Stability is analyzed using: an energy functional ensuring positivity operator inequalities bounding the coupling strength three stability regions (deep, mid, global) hybrid stability conditions derived from two-mode mixing Appendices provide operator-theoretic background, semiclassical expansions, and full proofs of the dispersion and stability results. Volume XXVII establishes the rigorous analytical backbone for computational and applied developments in subsequent volumes.