Information Protection as a Fundamental Principle IX: Rigorous Derivation of the Dirac Equation, the GUT-Scale Phase Transition, the Instanton Geometric Factor, and the Wilson Loop Threshold Coefficients

We complete the derivation of the Dirac equation from the discrete dynamics of plaquette defects on the internal SU(2) tensor network. The Clifford algebra emerges from the three mutually non-planar hopping paths on the Paley-9 graph, with the number of spatial dimensions fixed by the information-cost minimization theorem. We analyze the GUT-scale phase transition from first principles via the product graph spectrum, proving its first-order nature is robust. The instanton geometric factor on the Ramanujan-18 graph is computed as 1.769 from the complete spectral sum. The Wilson loop threshold correction coefficients are derived as (1/2, 1/3, 1/6) directly from the internal dimensions of the factor graphs, closing the last technical gap in the framework.