 Squared-population near-invariant for stochastic attrition

In the jump chain of the symmetric stochastic attrition process, the two possible increments of $D$ are $-2A+1$ and $2B-1$. Their mean is the displayed quantity, bounded by one in absolute value. Subtracting these conditional means produces a martingale. If both initial populations are at most $N$, it has at most $2N$ nonzero increments, each bounded by $4N$. Thus Azuma-Hoeffding inequality controls changes of order $N^2$ with exponentially small probability of order $e^{-cN}$.