I think the same arguments would translate to the torus and other translationally invariant spaces, so we could go higher dimensional as Mario suggested. Then I think one would need to replace $|S_{><}|$ by the minimal number $k$ such that there is a $C$ set for which $S\cup C$ is connected. I am not entirely sure how to generalise Lemma~\ref{lemma:probIndep} though, which has key importance in the present proof.

    Questions:

    \begin{itemize}

    	\item Is this proof finally flawless?

    	\item In view of this proof, can we better characterise $a_k^{(k+1)}$?

    	\item Why did Mario's and Tom's simulation show that for fixed $C$ the contribution coefficients have constant sign? Is it relevant for proving \ref{it:pos}-\ref{it:geq}?

    	\item Can we prove the conjectured formula for $a_k^{(3)}$?		

    \end{itemize} 

	\begin{lemma}On the infinite chain

	\end{lemma}   

    Note that using De Morgan's law and the inclusion-exclusion formula we can see that this is equivalent to saying:

    \begin{proof}

    We proceed by induction on $|I|$. For $|I|=0,1$ the statement is trivial.

    Now observe that:

    $$\P_I(Z^{0})=\sum_{P\text{ patch}\,:\,0\in P}\P_I(P\in\mathcal{P})$$

    $$\P_I(Z^{k})=\sum_{P\text{ patch}\,:\,k\in P}\P_I(P\in\mathcal{P})$$

    Suppose $|I|\geq 2$, then we proceed using induction similarly to the above

    \begin{align*}

    &\P_I(Z^{0}\cap Z^{k})\\

    &=\sum_{\underset{P_l^{\max}+1<P_r^{\min}}{P_l,P_r\text{ patches}\,:\,0\in P_l,k\in P_r}}

    \P_I(P_l,P_r\in\mathcal{P})