Emergence of the Livolsi Constant E⋆ and the Non Fundamental Nature of Physical Units from a Closed Quartic Variational Functional

This work presents a closed quartic variational framework in which all physical structures emerge from a single functional without external input. Starting from the variational equation, the analysis of the Hessian operator reveals a unique spectral structure characterized by a dimensionless invariant L=0.25L = 0.25L=0.25 associated with a minimal cyclic Z3Z_3Z3 closure. The full dynamics is generated by the pair (E⋆,L)(E_\star, L)(E⋆,L), where E⋆E_\starE⋆ arises as the principal eigenvalue of the Hessian and defines the global scale, while LLL encodes the internal geometric structure. A key result of the construction is that E⋆E_\starE⋆ is not independent, but is fully determined by LLL, leading to a complete reduction of all degrees of freedom. Physical constants such as ℏ\hbarℏ, ccc, and GGG are shown to emerge as composite quantities from an external mapping between internal variables and observer coordinates. Consequently, the International System of Units is not fundamental, but represents a human-defined parametrization of an underlying dimensionless structure. The framework admits no free parameters, no external constants, and no independent scales. All observables reduce to the form Physics=E⋆(L) Ln,\text{Physics} = E_\star(L)\, L^n,Physics=E⋆(L)Ln, demonstrating that all physical scales originate from a single structural invariant. This establishes a fully closed, self-sufficient, and non-modular variational system in which structure alone generates scale, dynamics, and measurable quantities.