RSW reflection extension lemma

For independent bond percolation on the square lattice at $p=1/2$, square crossing probabilities bounded below by self-duality extend to a uniform positive bound for rectangles of aspect ratio $3/2$. Expose an extremal vertical square crossing, reflect its geometry, and attach it to an exterior side through an unexposed independent region. Reflection symmetry bounds the attachment probability below. Two attachments and a horizontal square crossing join by the Harris-FKG inequality. Overlapping rectangles and transverse overlap-square crossings then give all fixed aspect ratios. Reflecting a geometrical graph path does not require its reflected edges to be open.

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.