A discrete energy scale is derived from a closed quartic variational functionalwithout the introduction of external parameters or phenomenological inputs.Starting from the exact functional and its full static equation, the Hessianstructure is constructed through explicit variation and spectral decomposition. The cyclic symmetry induced by the interaction defines a fundamental invariantsubspace, whose dimension is computed exactly via a projection operator.A minimal closure of this subspace, combined with the intrinsic spectraldegeneracy of the Hessian, yields a unique finite-dimensional configurationspace. 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 Livolsi constant L arises as the universal spectral ratio of the normalizedHessian operator, E* defines the global energy scale through the trace of theHessian, and ν determines the minimal unit of energy discretization: ΔE = E*/96. The derivation of L and E* from the internal structure of the functional ispresented in Appendix A, establishing a fully self-contained framework in whichstructure, scale, and quantization emerge from the same variational principle.
Livolsi Edoardo (Tue,) studied this question.