Quantum Gravity in the GTE/Phi_MDL Framework: Functional Completeness

We present a complete perturbative quantum-gravitational description of the GTE/Phi_MDL framework. The curved-background Lagrangian L[Phi_MDL; g_munu] is uniquely determined by Minimum Description Length (MDL) minimality and the Wald entropy consistency argument, with the non-minimal coupling forced to xi=0 by three independent arguments. The full nonlinear Einstein field equations G_munu = 8*pi*G*T_munu[Phi_MDL] follow as a derived consequence. UV finiteness on arbitrary smooth curved backgrounds is established via the DeWitt-Schwinger heat-kernel expansion and Hadamard propagator analysis: all curved-background-specific UV contributions reduce to finite renormalizations at the Planck scale, leaving no UV problem beyond flat spacetime. For black holes, the Hawking temperature T_H = M_Pl^2/(8*pi*M_BH) is unmodified by the Phi_MDL mass; the critical black-hole mass M_crit = M_Pl^2/(8*pi*m_tau) = 3.34e39 MeV is identified. Five equivalent descriptions of the GTE encoding structure — Lagrangian, algebraic/Reed-Solomon code, holographic/RT, MDL-information-theoretic, and QEC — are proved mutually equivalent in the GTE Holographic Encoding Theorem (established by derivation with all twenty directed implications closed). The Standard Model generation orbit is identified with a Reed-Solomon [5,3,3]_7 error-correcting code over GF(7), machine-certified in Lean 4 with zero sorry; the area unit a^2=4*l_Pl^2*log(7) is algebraically determined, not a free parameter. The MDL-minimal cosmological initial state — flat, field-kinetic-dominated, requiring log_2(3) ≈ 1.585 bits of initial data — dissolves the horizon, flatness, and domain-wall problems without inflation, and predicts a quantum bounce at Planck density. The primordial spectral index satisfies n_s = 1 exactly at leading order; a seven-step derivation chain yields the conditional prediction n_s = 1 - ln(2)/(2*pi^2) = 0.96488, matching observation at 0.004 sigma. The tensor-to-scalar ratio satisfies r=0, constituting the primary falsifiable prediction for the LiteBIRD satellite. The Galois group Gal(Q(zeta_7)/Q) is identified with CPT times generation-orbit symmetry, machine-certified in Lean 4 with zero sorry. A dimensionless structural prediction Omega_Lambda = (ln(2)/3*pi)*log_2(2000/3) ≈ 0.690 matches the observed dark-energy fraction at 0.70 sigma. Collectively, these results establish functional completeness across all six benchmark criteria of perturbative quantum gravity in the GTE/Phi_MDL framework.