= Closed-cut bound for percolation in a strip
A vertical cut between consecutive columns of $T_k$ contains $2k+1$ <edges>. All are closed with <probability> $(1-p)^{2k+1}$. Cuts in different columns use disjoint <edges>, so their closed-cut events are independent. A connection from column zero to column $n$ must cross each of the intervening $n$ cuts. Consequently
$$
q_k(n)\leq\bigl(1-(1-p)^{2k+1}\bigr)^n,\qquad f_k(p)\geq-\log\bigl(1-(1-p)^{2k+1}\bigr)>0.
$$
This finite-width obstruction persists even when the unrestricted planar percolation is supercritical.
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