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

This work develops a closed variational framework in which all physical structure emerges from a single quartic functional without external input. The analysis of the associated variational equation and its Hessian reveals a unique spectral organization governed by a dimensionless invariant L=0.25 L=0.25, arising from a minimal cyclic closure. A central result is that the global scale of the system, represented by E⋆ E ⋆ ​ , is not an independent parameter but is entirely determined by the structural invariant L L. This leads to a complete reduction of degrees of freedom: both structure and scale are expressions of a single underlying object. Within this framework, physical constants such as the Planck constant, the speed of light, and the gravitational constant are not fundamental, but emerge as composite quantities from a mapping between internal dimensionless variables and observer-defined coordinates. Consequently, the International System of Units is shown to be non-fundamental, functioning only as a representational layer. The resulting picture is one in which all physical observables reduce to functions of a single dimensionless invariant, establishing a fully self-contained, non-modular, and parameter-free structure. This suggests a reformulation of fundamental physics in which geometry alone generates scale, dynamics, and measurable quantities.