Complete Reduction of Physical Structure to the Livolsi Constants

We present a closed quartic variational framework in which all physical quantities emerge from a single functional without the introduction of external parameters, fitting procedures, or auxiliary theoretical structures. Starting from the variational principle δS[Ψ]δΨ†=0\frac{\delta S[\Psi]}{\delta \Psi^\dagger} = 0δΨ†δS[Ψ]=0 we obtain a unique admissible configuration whose second variation defines a finite-dimensional Hessian operator. Its spectral decomposition yields two emergent constants: the Livolsi constant L=0.25L = 0.25L=0.25 and the Livolsi constant E⋆E^\starE⋆, which together fully determine the structure of the theory. From this minimal spectral input, we derive a discrete hierarchy of energy levels En=E⋆LkE_n = E^\star L^kEn=E⋆Lk and show that all physical quantities—including the fine-structure constant α\alphaα, Planck constant ℏ\hbarℏ, speed of light ccc, gravitational constant GGG, and mass scales—emerge as necessary consequences of the internal structure of the functional. No fundamental constants are assumed. Physical units are shown to arise as relational quantities between spectral levels, and interactions emerge from coherence and deformation within the variational structure. The framework is fully closed, non-modular, and structurally rigid: any modification of the quartic functional destroys the spectral consistency and invalidates all derived quantities. This work establishes a self-contained mathematical structure in which physical reality is entirely determined by the spectral properties of a single variational principle.