= Solution
In <bond percolation> on the <square lattice>, every nearest-neighbour <edge> independently has state one, meaning open, with <probability> $p$, and state zero, meaning closed, with <probability> $1-p$. Thus the configuration law is the <product measure> $\mathbb P_p=\bigotimes_{e\in E}\operatorname{Bernoulli}(p)$. An <open path in bond percolation> uses only open <edges>. An <increasing event> $A$ satisfies $\omega\in A,\ \omega'\ge\omega\Rightarrow\omega'\in A$: opening further <edges> cannot destroy the event.
The <Harris-FKG inequality> for two increasing events is
$$
\boxed{\mathbb P_p(A\cap B)\ge\mathbb P_p(A)\mathbb P_p(B)}.
$$
For <disjoint occurrence of increasing events>, write $A\square B$ when disjoint sets of open <edges> separately witness $A$ and $B$, so each event is forced by its own witness regardless of the other <edges>. The <Van den Berg-Kesten inequality> states
$$
\boxed{\mathbb P_p(A\square B)\le\mathbb P_p(A)\mathbb P_p(B)}.
$$
For events given by <cylinder sets> these are finite-coordinate assertions. Connection events have finite open-path witnesses; truncating to finite boxes and taking increasing limits gives the same disjoint-occurrence bound for them. Sharing a <vertex> between the two witness paths is permitted; sharing an <edge> is not.
For $1\le m<n$, take a simple open path from the origin to the radius-$n$ boundary and let $x$ be its first <vertex> on the radius-$m$ boundary. Its initial segment witnesses $0\leftrightarrow x$. If $z$ is its final <vertex>, then $\|z-x\|_\infty\ge n-m$. Consequently the remaining segment reaches $x+\partial\Lambda_{n-m}$, and its initial portion until that visit witnesses $x\leftrightarrow x+\partial\Lambda_{n-m}$. The two segments use disjoint <edges>. Hence
$$
\{0\leftrightarrow\partial\Lambda_n\}
\subseteq\bigcup_{x\in\partial\Lambda_m}
\bigl(\{0\leftrightarrow x\}\square\{x\leftrightarrow x+\partial\Lambda_{n-m}\}\bigr).
$$
The <union bound>, <Van den Berg-Kesten inequality> and <translation invariance> of the <product measure> now give the <weighted BK boundary-splitting estimate>
$$
\boxed{g_n\le g_{n-m}\sum_{x\in\partial\Lambda_m}\mathbb P_p(0\leftrightarrow x)}.
$$
For $m=n$ the same bound follows directly from the <union bound>, with $g_0=1$. With the conventional $\partial\Lambda_0=\{0\}$, $m=0$ gives equality.
Write $C(0)$ for the <percolation cluster> of the origin. By summing its membership indicators, the <percolation susceptibility> is
$$
\chi(p)=\mathbb E_p|C(0)|=\sum_{x\in\mathbb Z^2}\mathbb P_p(0\leftrightarrow x)
=1+\sum_{m\ge1}a_m,\qquad
a_m=\sum_{x\in\partial\Lambda_m}\mathbb P_p(0\leftrightarrow x).
$$
If it is finite, then $a_m\to0$. For $0<p<1$, choose $m$ with $0<a_m=a<1$. Iterating the boundary-splitting estimate with this fixed $m$ gives
$$
g_k\le a^{\lfloor k/m\rfloor}g_{k-m\lfloor k/m\rfloor}
\le a^{\lfloor k/m\rfloor}.
$$
For $k\ge2m$, $\lfloor k/m\rfloor\ge k/(2m)$, so $g_k\le\exp[-(-\log a)k/(2m)]$. To handle all smaller positive radii without an unspecified prefactor, observe that $g_k\le g_1=1-(1-p)^4<1$. Therefore
$$
\boxed{\gamma=\min\left\{\frac{-\log a}{2m},\frac{-\log g_1}{2m}\right\}>0,
\qquad g_k\le e^{-\gamma k}\quad(k\ge0)}.
$$
For $1\le k<2m$, the second chosen rate gives $e^{-\gamma k}\ge g_1$, and for larger $k$ the first rate applies. At $k=0$ both sides equal one. The endpoint $p=0$ has $g_k=0$ for every $k\ge1$, so any positive rate works; $p=1$ is excluded by finite susceptibility. This proves <exponential one-arm decay from finite susceptibility> with the required unit prefactor.
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