Spacetime Granularity and Quantum Gravity: Construction of a Theoretical Model toward Ultraviolet Completeness

Based on the first principles of ontology, the minimum granularity of spacetime can itself serve as the ultraviolet cutoff of quantum gravity. Building on this insight, this paper completes a rigorous, end-to-end treatment of the foundational theory. Starting from the operational definition of spacetime granularity, we promote the minimum resolvable length $dx$ to a Lorentz scalar field φ=1/dx (with lower bound φP=1/lP), select the exponential potential V(φ)=VP e-φ/φP under five physical constraints, for the metric, for the dark energy field D, for the granularity field φ, and for the matter field ψ, write the complete action S[g,ψ,D,φ] and derive all the field equations---the absence of φ--matter coupling avoids a fifth force, and the low-energy limit returns naturally to the Einstein equations with a cosmological constant. We complete the ADM decomposition and the canonical quantization, obtain the Wheeler--DeWitt equation, identify the problem of time and propose a D--φ two-internal-clock proposal, treat the $f(D)$ nonminimal coupling in full, and prove background independence. On ultraviolet completeness: we recast the ultraviolet cutoff in the Lorentz-covariant form Λeff(x)=φ(x) and prove its covariance; compute the degree of divergence of gravitational loop diagrams, $D=2L+2$, and prove that under the pointwise covariant cutoff all loop diagrams are finite in magnitude, with IL~φ2L+2; write the renormalization group equation, find the non-Gaussian fixed point g_*=2(4π)²/c₁ in the one-loop approximation, and give preliminary evidence for asymptotic safety (a complete proof is left to future work). This paper states plainly: gravity is nonrenormalizable, but it can be treated as an effective field theory, with domain of validity EΛeff(x)=φ(x); all conclusions are confined to what has been proved, with no extrapolation beyond the evidence.