Gelation as a Dynamical Instability of the Smoluchowski Flow

together with the classical Smoluchowski, we show that gelation is consistently preceded by the appearance of positive real eigenvalues, indicating a loss of local dynamical stability. While nongelling kernels exhibit only transient finite-size effects, gelling kernels display persistent spectral destabilization associated with macroscopic gel formation. These results place gelation within a unified dynamical framework of aggregation-driven phase transitions by identifying a spectral signature that links the kinetic runaway of gelation to the metastability of nucleation in Becker-Döring kinetics.