Finite sectional coagulation-fragmentation balance with reported overflow

Input a grid $a=x_0<\cdots <x_M=R$, a nonnegative grid population $n_i$, a nonnegative reservoir mass $L$, symmetric nonnegative aggregation rates $K_{ij}$, nonnegative parent fragmentation rates ${\beta }_i$, and daughter-count measures $B_{x_i}$ supported in $[a,x_i]$ with $\int z{B}_{x_i}(dz)=x_i$. For each ordered pair $(i,j)$ use event rate $K_{ij}{n}_i{n}_j\mathrm{/}2$. If $x_i+x_j\le R$, remove one particle from each parent slot and add the fixed-pivot measure $P_{\mathrm{\Delta }}(x_i+x_j)$; if $x_i+x_j>R$, remove both parents and add mass $x_i+x_j$ to $L$, with no retained last-bin particle. For each parent $i$, use fragmentation rate ${\beta }_i{n}_i$, remove its parent particle, and add $\int P_{\mathrm{\Delta }}(z)B_{x_i}(dz)$ to the grid population.

The resulting finite-dimensional autonomous ODE is the sum of these event rates times their stated increments, with $\dot{L}$ equal to the total overflow-mass rate. Invalid grids, negative input populations, or a daughter measure with support outside $[a,x_i]$ return invalid-input. The reservoir is diagnostic truncation mass and has no later fragmentation or coagulation law.