Curvature Crystallization: A Theory of Semantic Ricci Flow

We develop a rigorous mathematical framework for Semantic Ricci Flow on k-NN graphs with Forman-Ricci curvature. Building on classical work by Hamilton and Perelman, we prove six main theorems: (1) Spectral Admissibility characterizing learnable curvature spectra, (2) Stability Threshold establishing τ > 0.88 as the critical stability condition, (3) Bimodal Convergence proving mode separation under flow, (4) Forbidden Surgery showing topological conservation, (5) Perelman-STL Connection linking semantic and geometric flows, and (6) Nonlinear Curvature Evolution deriving the exact form dκF /dt = γ1κ · davg + γ2Lw[κ] + γ3Ceκ with γ3 = 2η from first principles via discrete BBGKY closure. All results include complete proofs with no logical gaps.