R-layer Mode Theory (RLMT) LVI: Quantum Observables in R-layer Mode Theory

Volume LVI develops the quantum observables of R-layer Mode Theory (RLMT). Building on the quantum structures of Volume LV—including the algebra AQ, Hilbert space HQ, tension operator E, modular theory, and spectral triples—this volume formulates observables as operator–state pairs and analyzes: spectral observables of E and the modular Hamiltonian KQ, information-geometric observables such as quantum relative entropy and Connes distance, categorical observables from the oo-category of quantum layer transitions, spectral-geometric observables from the spectral triple, cosmological and holographic observables relevant to RLMT. Together with Volumes XLI-I, LIV, and LV, this work completes the observable-level structure of RLMT.