Exponential one-arm decay from finite susceptibility ($\chi (p)<\infty ⟹g_n\le e^{-cn}$)

The percolation susceptibility is $\chi (p)=1+{\sum }_{m\ge 1}{\sum }_{y\in \partial {\Lambda }_m}{\tau }_p(0,y)$. Finiteness gives a boundary sum $a<1$. The weighted BK boundary-splitting estimate then gives $g_k\le a^{\lfloor k/m\rfloor }$. The strict bound $g_1=1-(1-p{)}^{2d}<1$ absorbs the finitely many smaller radii into a positive exponential rate with unit prefactor. At $p=0$, all positive-radius connection probabilities are zero.