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

We show that a quark-like triplet structure of the nucleon can emerge directly from the spectral properties of a closed quartic variational functional. Starting from the stationarity condition of the functional, a perturbative expansion around the stationary configuration generates a Hessian operator whose structure is determined by a cyclic three-index contraction. This interaction produces a natural Z3Z_3Z3 symmetry in the perturbation space and leads to a circulant Hessian spectrum Spec(H)={1,L,L}\mathrm{Spec}(H)=\{1,L,L\}Spec(H)={1,L,L}, where the structural constant L=0.25L=0.25L=0.25 arises intrinsically from the variational dynamics. The resulting spectrum predicts a fixed internal energy gap ΔE=0.75E⋆\Delta E = 0.75E_\starΔE=0.75E⋆ and a discrete excitation hierarchy En=E⋆(1+nL)E_n = E_\star(1+nL)En=E⋆(1+nL). Using the emergent spatial scale of the stationary configuration, the model reproduces the characteristic nucleon energy scale E⋆≈939 MeVE_\star \approx 939\,\mathrm{MeV}E⋆≈939MeV, a quark-like internal energy scale Eq≈313 MeVE_q \approx 313\,\mathrm{MeV}Eq≈313MeV, and a nucleon radius rN≈0.84 fmr_N \approx 0.84\,\mathrm{fm}rN≈0.84fm. We further show that the triplet configuration is the minimal spectrally stable structure supported by the functional: configurations with p=1p=1p=1 are trivial, p=2p=2p=2 are unstable, while p=3p=3p=3 provides the unique minimal stable solution. Within this framework the nucleon appears as a coherent cyclic oscillation of three dynamically coupled internal modes. The familiar triplet structure usually attributed to constituent quarks therefore emerges naturally from the spectral properties of the underlying variational dynamics.