The Universal Spectral Gap

This work shows that the quartic variational framework with internal SU (2) symmetry necessarily produces a universal and strictly positive spectral gap at its stationary configuration. The Hessian at the global minimum has a fixed two-dimensional kernel determined by symmetry, while all other admissible directions form a finite-dimensional space, ensuring a fully discrete and isolated spectrum. The first nonzero eigenvalue, defined as Δₘin, is simple, unique, and independent of all stabilizing parameters of the functional. The torsional mode is the only admissible direction not removed by normalization and core constraints. Its frequency derives from the balance between the global quartic interaction and the internal Γ-trace term, leading to the universal scaling ωₜor = 3 α^ (3/2). No external potentials, geometric assumptions, or model-dependent inputs are required. The constant Δₘin therefore emerges as a genuine invariant of the theory: parameter-free, coordinate-independent, and unavoidable within the quartic variational structure.