Overtone Spectrum of the 47 MeV φ Resonance in the SGCU Framework: Derivation, Structure, and Experimental Signatures (Version 5.0)

The SGCU (Cosmic Topology) theory predicts a neutral scalar boson of mass m_φ ≈ 47 MeV, the φ resonance, as the ground-state quantum of normal vibration of spatial layers. Starting from the soliton geometry of SGCU, we derive the complete overtone spectrum of this particle, including radial overtones (n=1,2,3,), surface angular overtones (l=0,1,2,), and combined overtones from tangential winding and normal vibration. The core results are: the bulk fundamental mode at 47 MeV, the surface fundamental mode at 47/√8 ≈ 16.6 MeV (consistent with the ATOMKI X17 anomaly), the first radial overtone at approximately 94 MeV (including anharmonic corrections), surface angular overtones in the 16.6--17.0 MeV interval, and combined overtones such as the φ_μ resonance at approximately 153 MeV (with coupling shifts). All overtones are scalar bosons ($C=0, Q=0$), with low-order overtones having widths on the keV scale. We explicitly distinguish strict analytical results from phenomenological estimates, provide anharmonic corrections for radial overtones, clarify the spin-parity of angular modes, give coupling shifts for combined overtones, analyze the stability and width of higher overtones, and summarize existing experimental constraints in each mass window. The experimental verification of the overtone spectrum---a "spectroscopic" pattern of narrow scalar resonances---would constitute decisive evidence for the SGCU geometric unification framework, analogous to the role of atomic spectroscopy for the Bohr model.