Emergent Quark-Like Internal Structure of the Nucleon from the Ψ–Γ Quartic Functional

We investigate whether the three-component internal structure commonly attributed to quarks in the nucleon can emerge directly from the spectral properties of a closed variational system. Starting from a quartic variational functional, we analyze the stationary configuration and the spectrum of perturbations generated by its Hessian operator. The structure of the interaction term induces a cyclic coupling among three internal modes, leading naturally to a Z3 Z 3 ​ symmetry in the perturbation space. The Hessian operator therefore takes a circulant form whose spectrum is Spec(H)={1,L,L} Spec(H)={1,L,L}, where the structural constant L=0.25 L=0.25 arises intrinsically from the variational dynamics. We show that the triplet configuration is the minimal spectrally stable structure supported by the functional: configurations with one or two components are either trivial or unstable, whereas the three-component configuration forms the unique minimal stable solution. Within this framework the nucleon appears as a coherent cyclic oscillation of three dynamically coupled internal modes. The familiar quark-like triplet structure therefore emerges as a consequence of the spectral properties of the underlying variational dynamics rather than as a postulated set of elementary constituents.