Reflected attachment estimate for percolation (${\mathbb{P}}_p(X)\ge \frac{1}{2}{h}_p(m,2n)h_p(n,n)$)

In a rectangle $[0,m]\times [0,2n]$, $m\ge n$, let $X$ require an open vertical crossing of the corner square $[0,n{]}^2$ attached to the rectangle's right side. Explore the leftmost vertical crossing of the square without inspecting edges to its right. Reflect its geometry in the line $y=n$, obtaining a deterministic separator. Every horizontal rectangle crossing reaches this separator from the right; reflection divides these possible attachments into two equally probable choices, so the lower-half attachment has probability at least half the horizontal crossing probability. The required edges remain unexposed and independent of the explored crossing. Averaging proves the estimate. Reflected geometry need not be open.