R-layer Mode Theory (RLMT) Appendix V: Unified Spectral Foundations and Eigenvalue Hierarchy

Appendix V develops the unified spectral foundations of R-layer Mode Theory (RLMT) by establishing the mathematical structure of the tension operator E and its eigenvalue hierarchy λn. We construct the self-adjoint realization of E, derive its spectral resolution, and analyze the asymptotic behaviour of the eigenvalues through Weyl-type laws, spectral densities, and gap scalings. These results provide a rigorous interpretation of the RLMT layer hierarchy and clarify how different physical sectors correspond to distinct projections onto the spectrum of E. As an application, we analyze the massive cold molecular gas reservoir in the galaxy REBELS‑25 at z=7. 31. The disc radius, gas mass, and rotation curve are shown to be consistent with the RLMT hierarchical index α≈3, indicating that the layered tension hierarchy is already established by this early epoch. This appendix serves as a foundational reference for all RLMT volumes—observational, geometric, quantum, informational, and cosmological—by providing the unified spectral language through which layered tension, hierarchical indices, and multi-sector observables acquire a coherent interpretation.