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 Z3Z₃Z3 symmetry in the perturbation space and leads to a circulant Hessian spectrum Spec (H) =1, L, LSpec (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⋆ E = 0. 75E_ΔE=0. 75E⋆ and a discrete excitation hierarchy En=E⋆ (1+nL) Eₙ = E_ (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_ 939\, MeVE⋆≈939MeV, a quark-like internal energy scale Eq≈313 MeVEq 313\, MeVEq≈313MeV, and a nucleon radius rN≈0. 84 fmrN 0. 84\, fmrN≈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.