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

This paper investigates the emergence of exotic hadronic states within a closed quartic variational framework. The physical configurations of the system are determined by the stationary solutions of a globally defined quartic functional. The second variation of this functional generates a fluctuation operator whose spectral structure determines the dynamical modes, energy levels, and mass spectrum of localized excitations. The quartic interaction contains a cyclic three-index contraction that induces a non-trivial internal spectral organization. This structure produces a minimal spectral sector consisting of a cyclic triplet together with a singlet mode, generating a natural (3+1) hierarchy of resonances. The resulting eigenvalues determine a discrete mass spectrum without the introduction of phenomenological parameters or external fitting. From the spectral structure of the fluctuation operator, numerical values for the lowest resonant modes are obtained. These values fall in the same mass region as several exotic hadronic states observed in high-energy experiments at CERN, including well-known tetraquark candidates. The predicted ordering of the spectrum reproduces the clustering of resonances reported in experimental analyses. The comparison suggests that the observed exotic hadrons may reflect an underlying spectral organization generated by the quartic variational structure. The results therefore indicate that tetraquark-like states can emerge naturally from the internal dynamics of a closed variational system. More precise experimental determinations of resonance masses could provide a direct test of the spectral hierarchy predicted by the framework.