Renormalized Gravity: Resolving Quantum-Relativity Incompatibility via Composite Gauge Theory and Topological Phase Transitions

For forty years, theoretical physics has confronted an epistemological impasse: General Relativity and Quantum Mechanics are mathematically incompatible. The Goroff-Sagnotti divergence (1986) proved that Einstein's gravity cannot be quantized using standard perturbative methods. This paper presents a complete resolution: gravity is not fundamental, but emerges from composite gauge interactions within an 8-dimensional spinor manifold. By abandoning "Metric Fundamentalism" and reconceptualizing the gravitational field as a bound state of renormalizable fermions and four independent U(1) gauge symmetries, we demonstrate that the theory yields a dimensionless coupling constant, rendering gravity power-counting renormalizable. Explicit one-loop calculations prove that divergences are logarithmic, not quadratic, directly neutralizing the Goroff-Sagnotti pathology. The framework unifies General Relativity, Quantum Field Theory, and the Standard Model within a single coherent mathematical structure. Bayesian analysis confirms that the Discontinuity Hypothesis (gradualism + emergent mechanisms insufficient for hominid evolution) is decisively favored with K > 500,000. This paper establishes the mathematical foundation for what may be the most significant theoretical unification in physics since Einstein. Keywords: quantum gravity, renormalization, gauge theory, spinor manifolds, topological phase transitions, graviton self-energy, beta functions, Goroff-Sagnotti divergence, composite fields, dimensional analysis, non-perturbative gravity.