Volume XLI‑R develops the rigorous spectral foundations of the tension operator E, which defines the layered structure of R-layer Mode Theory (RLMT). We construct the unique self-adjoint realization of E, derive its spectral resolution, and formulate both discrete and continuous layer hierarchies using projection-valued measures. This volume provides the mathematical basis for all observational and theoretical developments across the XLI series. As an application, we analyze the massive cold molecular gas reservoir in the galaxy REBELS‑25 at z=7.31. Using recent VLA and ALMA observations, we show that the disc radius, gas mass, and rotation curve are consistent with the RLMT hierarchical index α≈3, previously inferred from local disc galaxies. The detection of low‑J CO emission against the bright CMB at this epoch constrains tension‑induced corrections to radiative transfer. This volume serves as the mathematical anchor of the XLI series, demonstrating that the layered tension hierarchy is already established by z∼7 and that early massive discs are natural probes of the RLMT framework.
Tsuyoshi Tohi (Tue,) studied this question.