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.
Tsuyoshi Tohi (Sun,) studied this question.