Emergence of the 3 Livolsi Constant L=0.25, E, v=1/96

A discrete energy structure is derived from a closed quartic variational functional without the introduction of external parameters or phenomenological assumptions. Starting from the exact functional and its full variational equation, the Hessian operator is constructed explicitly and analyzed through its spectral decomposition. The cyclic symmetry induced by the interaction defines a fundamental invariant subspace, whose dimension is obtained exactly via a projection operator. A minimal closure of this subspace, combined with the intrinsic spectral degeneracy of the Hessian, yields a unique finite-dimensional configuration space. This construction leads to the emergence of three fundamental structural constants, denoted as the Livolsi constants: L = 0.25, E*, and ν = 1/96. The constant L arises as a universal spectral ratio of the normalized Hessian, E* defines the global energy scale, and ν determines the minimal unit of energy discretization. The resulting framework provides a self-contained structural mechanism in which symmetry, scale, and quantization emerge from the same variational principle, without external input.