The Uniqueness of the Ψ–γ Variational Structure

We consider the class of autosufficient, globally variational functionals depending exclusively on a complex field and its adjoint, subject to the minimal structural constraints of normalization stability, non-trivial interaction, spectral stability, internal closure, and absence of external structures. Within this class, we perform a complete and explicit structural analysis. We prove that any admissible functional must necessarily exhibit a quartic interaction as the minimal non-trivial degree, develop a non-local operator to avoid pointwise degeneracy, and admit a multi-index internal structure generating a finite, non-degenerate Hessian spectrum. Spectral stability uniquely fixes the structure to a three-component configuration with irreducible cyclic closure of order three. All alternative constructions are explicitly excluded by direct violation of the defining conditions. As a result, the admissible functional is uniquely determined up to internal isomorphism and reduces to a single closed quartic variational structure. The consequence is structural rigidity: no deformation, extension, or reduction preserves the defining conditions. Within the admissible class, no alternative functional exists.