The Livolsi Structural Constant L=0.25 and the Hierarchy of Physical Scales

This work investigates the consequences of a quartic variational functional generating stable cyclic configurations of internal fields. The stationary solutions of the functional are analyzed through the spectral properties of the Hessian operator. The minimal stable configuration is shown to correspond to a cyclic triplet structure satisfying the closure relation $\Gamma^3 = I$, producing the discrete symmetry group $Z_3$. The spectral analysis of the Hessian reveals a characteristic eigenvalue structure ${1, L, L}$ which defines a structural constant $L = 0.25$. This constant organizes a hierarchy of fluctuation modes and generates scale relations across multiple physical domains. The triplet structure produces internal cyclic currents which naturally lead to interference patterns in elastic proton--proton scattering. Observable quantities such as the elastic slope, the proton radius, and the position of diffractive minima are shown to be compatible with measurements from the TOTEM experiment at the Large Hadron Collider. The same structural constant generates hierarchical relations connecting hadronic scales, baryonic excitation spectra, leptonic mass hierarchies, dimensionless interaction strengths, and large-scale physical quantities. In particular, successive powers of the constant produce relations linking the nucleon mass scale to the gravitational Planck scale and strongly suppressed energy densities consistent with cosmological vacuum energy observations. These results suggest that a single structural constant emerging from the spectral properties of the underlying variational functional may organize a wide range of physical scales through a unified hierarchical structure.