Critical-mass conjecture for discrete coagulation-fragmentation equations

Consider the discrete coagulation-fragmentation equation with multiplicative coagulation and constant fragmentation kernels, and let $\rho_0$ be initial data satisfying the paper's initial-data condition. Write

$$m=m_1(\rho_0)$$

for its initial mass. Critical-mass conjecture. The critical mass should be $m=1$: for $m>1$, gelation occurs, while for $0<m\leq 1$ the equation has a unique mass-conserving solution for all time. This conjecture identifies the threshold between gelation and global mass conservation; the formal moment calculation supports gelation above the threshold and suggests a borderline phenomenon at $m=1$.