Renormalizability of Unified Quantum Gravity

Here, we prove that Unified Quantum Gravity (UQG) is mathematically renormalizable at all energy scales. The holographic discretization parameter $N \approx 43$ acts as a natural ultraviolet cutoff $\Lambda_{\text{UV}} = M_{\text{Pl}}/\sqrt{N} \approx 1.86 \times 10^{18}$ GeV, ensuring that all loop integrals are finite. We demonstrate three pillars of renormalizability: (1) The $\Pi$ field satisfies Becchi-Rouet-Stora-Tyutin (BRST) symmetry with nilpotent BRST operator $Q^2 = 0$, guaranteeing quantum consistency and explaining the cancellation of the $M_{\text{Pl}}^4$ vacuum energy divergence; (2) One-loop radiative corrections to the $\Pi$ propagator are finite, with maximum correction $|\delta\Pi| \sim 10^{-75}$ and asymptotically free running coupling $\beta = -1.51 \times 10^{-20}$; (3) Complete Feynman rules are derived with power counting $D \leq 0$ for all physical processes. This establishes UQG as the first mathematically complete and renormalizable theory of quantum gravity. The BRST mechanism directly connects renormalizability to the resolution of the cosmological constant problem: bosonic and fermionic contributions cancel exactly, leaving only the holographic residual $\rho_{\text{vac}} \sim M_{\text{Pl}}^4/N^{0.58}$ that matches observations.