Exact self-dual rectangle crossing probability (${\mathbb{P}}_{1/2}(H_n)=1/2$)

For the lattice rectangle with graph vertices $\{0,\dots ,n\}\times \{0,\dots ,n-1\}$, let $H_n$ be an open left-to-right crossing. Its complementary event is a closed dual top-to-bottom crossing. The dual crossing rectangle has width $n-1$ and height $n$, so rotation and translation identify it with the original rectangle. Boundary edges along the starting and ending sides do not affect the crossing event. At $p=1/2$ the primal and dual edge states have identical laws, henceThe one-unit adjustment in the side lengths makes the symmetry exact rather than an informal assertion about a finite graph vertex square.