R-layer Mode Theory (RLMT) XLI‑H: Quantum‑Gravitational Layer and the Modular Origin of Spacetime

This volume XLI‑H completes the quantum closure of R-layer Mode Theory (RLMT) by promoting the tension operator E to a quantum operator acting on a non‑commutative operator algebra. Using the GNS representation and Tomita–Takesaki modular theory, we construct the quantum‑gravitational layer RQG, defined by the deepest part of the spectrum of the quantum tension operator. The modular flow generated by EQG yields an emergent notion of time, while the spectral structure determines the emergent spatial geometry. Quantum fluctuations of the tension operator induce universal Planck‑scale corrections to gravitational waves, high‑energy neutrinos, and electromagnetic cosmology. These corrections lead to correlated multi‑messenger signatures and provide explicit falsifiability criteria for the theory. XLI‑H establishes a unified operator‑algebraic origin for the layered structure of spacetime and forms the quantum foundation of the entire R‑layer program.