Emergence of the Electron Mass from the Spectral Minimum of a Closed Quartic Functional

This work demonstrates that a strictly positive and unavoidable spectral gap emerges from a closed quartic variational structure. The framework is entirely defined by four unreduced equations: the universal quartic functional, its static variational equation, the corresponding dynamical equation, and a global decisional operator that selects the coherent solution among all admissible configurations. The analysis shows that the Hessian of the quartic functional at the globally admissible configuration possesses an exact two–dimensional kernel generated by the internal operator Γ. All remaining directions acquire strictly positive curvature, forcing the existence of a nonzero first eigenvalue. This quantity defines a universal spectral constant, denoted Δ_min, which represents the smallest admissible excitation of the system. The spectral gap is shown to be structurally unavoidable: it follows solely from the quartic interaction, the internal algebraic closure induced by Γ, and the global admissibility enforced by the decisional operator. No external parameters, background geometries, renormalization procedures, or phenomenological inputs are required. The result implies that discrete spectral structure, minimal oscillatory scales, and threshold excitations arise as necessary consequences of the variational framework. The universal spectral gap therefore constitutes a parameter-free invariant of the theory and acts as the fundamental seed from which emergent frequencies, masses, and transition scales originate. All derivations follow strictly from the full quartic functional and its associated variational equations, without importing structures from external physical theories.