Quantum Gravity in the GTE/Phi_MDL Framework: Functional Completeness

We establish perturbative quantum-gravitational completeness for the GTE/Phi_MDL framework across six benchmark criteria. The curved-background Lagrangian L[Phi_MDL; g_uv] is uniquely determined by MDL minimality and the Wald entropy argument, with xi = 0 forced by three independent arguments, and the Einstein field equations G_uv = 8 pi G T_uv[Phi_MDL] follow as a derived consequence. UV finiteness on arbitrary smooth curved backgrounds is established via DeWitt-Schwinger heat-kernel and Hadamard propagator analysis: curved-background UV contributions reduce to finite Planck-scale renormalizations, leaving no UV problem beyond flat spacetime. The GTE Holographic Encoding Theorem establishes the relationship among five descriptions of the GTE encoding structure (Lagrangian, Reed-Solomon code, holographic/RT, MDL, and QEC): twelve of the twenty directed implications, linking the Lagrangian, Reed-Solomon, MDL, and QEC descriptions to one another, are established by derivation with no gravitational input (unconditional CatAD); the remaining eight implications, which connect any of those four descriptions to the holographic/RT description, are established conditional on the standard Bekenstein-Hawking coefficient 1/4G taken as an external premise (conditional CatAD) — as in prior thermodynamic derivations of gravity, not as an output of the theorem. The SM generation orbit is a Reed-Solomon [5,3,3]_7 code over GF(7) (CatAL, zero sorry); given the external BH coefficient, the RS-code entropy per symbol fixes a consistent area unit a^2 = 4 l_Pl^2 log 7. The MDL-minimal initial state (flat, field-kinetic-dominated, log_2 3 approximately 1.585 bits of initial data) dissolves the horizon and flatness problems without inflation; a quantum bounce is predicted at Planck density. The domain-wall problem is resolved dynamically: the transient Z_7 wall network formed at the ordering crossover T_G approximately 0.70 GeV is annihilated by the canonical coupling bias well before nucleosynthesis, leaving no surviving defect network. The conditional prediction n_s = 1 - ln(2)/(2 pi^2) = 0.96488 (-0.005 sigma from Planck 2018) requires the EFE-bridge step. The tensor-to-scalar ratio r = 0 is the primary falsifiable prediction for LiteBIRD. The dark-energy fraction Omega_Lambda = 0.6899 matches observation at +0.18 sigma from Planck 2018 (conjecture; quantum mechanism open).