Emergence of Exotic Hadronic States, Mass Spectra and Resonance Frequencies (CERN)

This work investigates the emergence of exotic hadronic states within a closed quartic variational framework. Starting from a globally defined quartic functional, the physical configurations of the system are selected through the variational equation and the associated fluctuation operator. The second variation of the functional defines a Hessian operator whose spectral structure determines the dynamical frequencies, energy levels, and mass spectrum of localized excitations. The quartic interaction term generates a non-trivial cyclic structure associated with the triple index contraction appearing in the functional. This structure induces a minimal spectral organization characterized by a threefold cyclic sector together with a singlet mode, producing a natural (3+1) spectral block. The resulting eigenvalues determine a discrete hierarchy of resonant modes without the introduction of external parameters or phenomenological fitting. From the spectrum of the fluctuation operator, a set of emergent mass values for the lowest spectral modes is obtained. These predicted masses fall in the same mass region as several exotic hadronic resonances observed in high-energy experiments at CERN, including well-known tetraquark candidates. The predicted spectral ordering reproduces the clustering of these resonances and suggests that their structure may reflect the intrinsic spectral organization generated by the quartic variational functional. The comparison with experimental measurements indicates that the theoretical spectrum lies very close to the currently reported resonance masses. The remaining differences are small and appear to be comparable to the present experimental uncertainties and resonance widths. This suggests that improved precision in future measurements could provide a decisive test of the spectral structure predicted by the variational framework. The results presented here indicate that exotic hadronic states may arise naturally from the spectral properties of a closed quartic variational system. More precise experimental analyses in the corresponding mass region could therefore play an important role in clarifying whether the observed resonances follow the discrete hierarchy predicted by the theory.