Square-lattice even graphs and parity contour separation

Around each dual vertex the product of the four primal nearest-neighbor spin products is $1$, since each corner spin occurs twice. Hence the number of disagreeing bonds incident to that vertex is even. The mismatch graph is finite because all spins outside the finite box are $+1$.

Start from an unused edge and extend a trail along unused edges. Since every vertex has even degree, the trail cannot stop at a vertex other than its starting vertex; finiteness therefore gives a closed trail. Taking the segment between the first repeated vertex extracts a simple cycle. Removing its edges preserves even degree at every vertex, so repeating the finite procedure decomposes every edge into edge-disjoint cycles. The square lattice has no loops, parallel edges, or triangles, so each cycle has length at least four.

Dual vertices have half-integer coordinates, so a ray from an integer-height primal site meets no dual vertex. For horizontally adjacent sites, the symmetric difference of their rightward rays is exactly the horizontal unit segment crossing their joining primal bond, so their parity differs exactly when that bond is crossed by $\gamma$. For vertically adjacent sites $u=(i,j)$ and $v=(i,j+1)$, choose an integer $M$ larger than every horizontal coordinate of $\gamma$ and consider the rectangle $[i,M]\times [j,j+1]$. Its boundary contains no dual vertices. As the closed cycle is traversed, the indicator of whether its current vertex lies inside this rectangle toggles at each transverse boundary crossing and returns to its starting value, so the total number of boundary crossings is even. There is no crossing on the far side $x=M$; the other crossings are exactly those of the two truncated horizontal rays from $u,v$ and the dual edge across the primal bond $\{u,v\}$. Hence the two ray-intersection counts have different parity exactly when that vertical bond is crossed by $\gamma$.

Every primal site outside a rectangle containing $\gamma$ has parity zero, since its ray either misses $\gamma$ or crosses the full cycle an even number of times; thus the inside is finite. In a plus-boundary box, any site outside the box has an outward lattice path to infinity lying entirely in the plus exterior. Along this path no bond is a mismatch, so step 3.1 keeps its parity zero; hence every cycle-inside site lies in the box. Along any primal path, the spin changes exactly on mismatch edges. A path from the origin to the all-plus exterior therefore crosses an odd number of mismatch edges if $s_0=-1$. For each cycle in the edge-disjoint decomposition from step 2.1, step 3.1 says its parity changes along the path exactly when that path crosses one of its edges; its parity is zero at the exterior endpoint. The sum modulo two of the cycle parity changes is the total mismatch-edge count, which is odd, so at least one cycle has origin parity one, as claimed.