Emergence of the Gravitational Constant from a Closed Quartic Variational Functional

This work derives the gravitational constant from the spectral structure of a closed quartic variational functional. The analysis begins with a self-contained quartic functional whose stationary configurations define the variational vacuum of the framework. A fluctuation expansion around the stationary solution generates a Hessian operator whose spectrum governs the dynamics of perturbations. The continuous branch of the spectrum exhibits a quadratic dispersion relation λ(k)∼k2\lambda(k)\sim k^2λ(k)∼k2 which produces a propagator with radial dependence K(r)=14πr.K(r)=\frac{1}{4\pi r}.K(r)=4πr1. This structure generates a long-range interaction identical to the Newtonian gravitational potential. The coupling strength of the interaction is controlled by the spectral gap E⋆E_\starE⋆ of the Hessian. Combining the spectral scales emerging from the variational framework yields the relation G=14ℏcE⋆2.G=\frac{1}{4}\frac{\hbar c}{E_\star^2}.G=41E⋆2ℏc. Using the observed gravitational constant determines the spectral gap E⋆≈6.8×101 GeV.E_\star \approx 6.8\times10^{1}\,\mathrm{GeV}.E⋆≈6.8×101GeV. The derivation shows that the gravitational constant can be interpreted as a quantity determined by the spectral properties of a closed quartic variational functional rather than as a fundamental empirical parameter. The resulting framework connects the structure of the variational functional, the Hessian spectrum, the universal spectral gap, and the gravitational coupling.