Ontology V8.4.1:From Spacetime Granularity to the Complete Action

Building on Ontology V8.4, this paper completes the key transition from a kinematic formula to a dynamical theory. The main results are: 1. promoting the spacetime granularity \(dx\) to a scalar field \(\phi=1/dx\), with dimension \([L^{-1}]\), whose physical identity is the inverse-length scale field of spacetime granularity;2. writing the kinetic and potential terms of \(\phi\), and obtaining the Lagrangian density \(\mathcal{L}_\phi=\frac12(\partial\phi)^2-V_Pe^{-\phi/\phi_P}\);3. writing the complete action \(S[g,\psi,D,\phi]\), consisting of gravity, the \(D\) field, the \(\phi\) field, and matter;4. performing the ADM decomposition, obtaining the Hamiltonian constraint \(\mathcal{H}=0\) and the momentum constraint \(\mathcal{H}_i=0\);5. performing canonical quantization, obtaining the Wheeler-DeWitt equation \(\hat{\mathcal{H}}\Psi=0\);6. identifying the problem of time and proposing a dual internal-clock scheme based on the \(D\) field and the \(\phi\) field;7. fully treating the non-minimal \(f(D)\) coupling and proving background independence. This paper makes clear that the \(\phi\) field is the scalar field of the spacetime ontology: it decouples in the low-energy limit and becomes significant at the Planck scale. \(\phi\) has a lower bound \(\phi_P=1/l_P\), corresponding to an upper bound \(l_P\) on the spacetime granularity. The time quantization \(dt=t_P\) remains an additional assumption, but it is already connected to continuous time through the internal-clock scheme. \textbf{Keywords:} spacetime granularity; scalar field \(\phi\); complete action; ADM decomposition; Wheeler-DeWitt equation; problem of time; internal clock; background independence