A convexity-type invariant for the critical coagulation--fragmentation Hamilton--Jacobi equation

We study the critical coagulation--fragmentation equation with multiplicative coagulation kernel $a(s,\hat s)=s\hat s$ and constant fragmentation kernel $b(s,\hat s)=1$. Under the Bernstein transform, mass-conserving solutions correspond to solutions of a singular Hamilton--Jacobi equation studied by Tran and Van (Comm.Pure Appl.Math.75 (2022), no.6, 1292--1331). Through this correspondence they proved that mass-conserving solutions are unique on the full critical range $0<m\le1$, but could establish their existence only for $0<m<\tfrac12$. We identify a one-sided, convexity-type invariant that holds for Bernstein-transform data and is propagated by their viscous scheme as a genuine maximum-principle bound. We call it the half-slope invariant. It sharpens the curvature barrier and thereby extends mass-conserving existence to the entire critical range $0<m\le1$. Hence $m=1$ is the critical mass, confirming the threshold predicted by Vigil and Ziff (J.Colloid Interface Sci.133 (1989), no.1, 257--264). The same invariant appears in the radial partial-mass formulation of the two-dimensional Keller--Segel equation, whose critical mass is $8\pi$.