Emergent Quantization and Gravity from a Fractional PDE on the Sephirotic Graph Topology

We demonstrate that quantization and gravity emerge simultaneously from a single mathematical structure: an 11-field coupled PDE system with fractional Laplacian (-Δ)^{α/2} defined on the Sephirotic graph G₁₁. The fractional index α = 6/5, previously shown to be an exact eigenvalue of the normalized graph Laplacian (Paper 9), produces both a discrete energy spectrum (quantization without axioms) and a modified gravitational Green's function G_α(r) ~ r^{α-3} that naturally yields flat galactic rotation curves without dark matter. We prove that: (1) the fractional spectrum on G₁₁ generates 10 discrete energy levels with variance ratio 791.7 relative to equal spacing, confirming non-trivial quantization; (2) the Green's function exponent α - 3 = -1.8 transitions smoothly to the Newtonian -1 as α → 2; (3) the 11-field PDE system maintains bounded energy with 92% coherence over long evolution; (4) phase transitions in the spectral structure emerge at critical coupling strengths, providing a dynamical mechanism for meltdownFrac thresholds; (5) both standard quantum field theory (spectral distance → 0) and general relativity (gravitational deviation → 0) are recovered simultaneously in the α → 2 limit. This unification requires zero free parameters beyond the topological invariant α = 6/5, connecting all 11 previously demonstrated domains — from neutrino oscillations to qubit decoherence — through a single operator on a single graph.