Emergence of Neutrino Mass from the Structural Threshold Defined by the Livolsi Constant L

This work demonstrates that the neutrino mass arises as an unavoidable structural consequence of the quartic variational functional. Building on the emergence of the dimensionless Livolsi constant L=0.25L = 0.25L=0.25, we show that this invariant represents the minimal curvature required for the stability of the symmetric phase. When the cyclic sector approaches this threshold, the Hessian loses positivity and the system is forced into a new configuration. The first admissible departure from symmetry defines the smallest nonzero mass eigenvalue of the dynamic operator. This eigenmode corresponds to the neutrino, whose mass is therefore not a parameter, coupling, or phenomenological input, but the intrinsic result of variational closure. The analysis establishes a direct structural link between the stability threshold of the quartic functional and the origin of the neutrino mass, providing a non-tunable and universal explanation for why the neutrino cannot be massless and why its mass is naturally much smaller than other physical scales.