Ontology V8.4.2: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, and systematically treats ultraviolet completeness and the covariant cutoff. The main results are: (1) promoting the spacetime granularity dx to a scalar field φ=1/dx; (2) writing the kinetic and potential terms of φ and obtaining the Lagrangian density; (3) writing the complete action S[g,ψ,D,φ]; (4) performing the ADM decomposition to obtain the Hamiltonian and momentum constraints; (5) performing canonical quantization to obtain the Wheeler–DeWitt equation; (6) identifying the problem of time and proposing a dual internal-clock scheme; (7) fully treating the non-minimal f(D) coupling and proving background independence; (8) rewriting the ultraviolet cutoff Λ_eff=φ in a Lorentz-covariant form; (9) computing the gravitational loop divergence and showing that all loop diagrams are finite under the covariant cutoff; (10) writing the renormalization-group equation, finding a non-Gaussian fixed point, and showing asymptotic safety; (11) writing the higher-curvature operators, computing their running, showing they are negligible at low energy, and giving the regime of validity of the effective field theory. This paper makes clear that gravity is non-renormalizable but can be treated as an effective field theory; the covariant cutoff Λ_eff(x)=φ(x) yields finite loop diagrams; the ultraviolet is completed by asymptotic safety. φ has a lower bound φ_P=1/l_P. Keywords: spacetime granularity; scalar field φ; complete action; ADM decomposition; Wheeler–DeWitt equation; problem of time; internal clock; background independence; covariant cutoff; loop divergence; asymptotic safety; non-Gaussian fixed point; higher-curvature operators; effective field theory