The Structural Inevitability of the Livolsi Variational Functional

We establish that the closed quartic Psi–Gamma variational functional is not one possible model among alternatives, but the unique structure compatible with the existence of non-trivial, stable, and self-contained physical systems. Starting from minimal conditions of existence, stability, absence of external structure, and non-degenerate spectral behavior, we derive a complete set of admissibility constraints. We show that any variational functional satisfying these conditions must necessarily reduce, up to internal isomorphism, to a single form. All alternative constructions are excluded by structural failure, leading to degeneracy, instability, loss of closure, or external dependence. The result implies a collapse of the space of admissible theories to a singleton. The absence of alternatives is not assumed, but follows as a direct consequence of the admissibility conditions.