3.8 Quarkonium Spectrum: High-Dimensional Geometric Origin

Abstract

Starting from the {2,3,5} Interlock Property framework, we study the stable configurations of two particles on the joint Sector Parameter Space and the geometric origin of the quarkonium spectrum. On the joint Sector Parameter Space, the ℳ Sector effective Spectral Gap is , the Dirac spectrum retains , and the effective curvature ratio drops from 151.7 (single particle) to 75.85. An existence theorem for the minimum of the Algebraic Action on the joint Sector Parameter Space is proved: under the ℳ Sector synchronization constraint, the joint Sector Parameter Space is a convex submanifold, and the variational equation has a unique minimum solution. The effective mass is jointly determined by the synchronized ℳ Sector angle at the minimum of the joint action and the geometric factor of the joint Sector Parameter Space. Color singlet corresponds to equal-weight superposition in the ℐ Sector Complex Structure; non-color-singlet states are exponentially suppressed by the Dirac spectrum due to action divergence. The theoretical framework establishes correspondences with the inevitable structure of the quarkonium spectrum: Dirac spectral energy scale → rigid locking of deeply bound states; many-body spectral parallelism → tetraquark families; metastable saddle-point solutions → non-color-singlet narrow resonance spectra. **Keywords**: Quarkonium spectrum; Joint Sector Parameter Space; Spectral parallelism; Color singlet; {2,3,5} Interlock Property