Lemma 18 ([Preserving a connected zero exterior]). Let D_n be a finite simply connected triangular-lattice polygon. Let E_n consist of its boundary vertices and all vertices outside its interior, and put u=0 on E_n . Assume u is uniformly …

Use the original lattice vertices for the extension. Bad boxes are first enlarged by the fixed amount needed to control every original edge on a surrounding good rim. Their outer filled hulls give the holes of Lemma 5 ([Sparse-hole extension]); passing from boxes to their unions of lattice sites changes diameter and counting constants only by the fixed box size. Enlarge the rim by a fixed layer if needed to include an exit by a triangular-lattice diagonal edge. The same square-coordinate contour construction supplies connected good rim paths of length at most C(d+1)^2 for a hole of diameter d . No extra probabilistic claim for independently marked individual sites is needed. The zero set E_n is connected to infinity in the lattice graph. Indeed the polygon has one connected boundary cycle, and its exterior is the unbounded complementary component; including boundary vertices removes any vertex-level corner separation. For a hole H which meets E_n , follow an E_n -path from a vertex in H to infinity. Since H is finite, the path exits it. Its first exterior rim vertex lies in E_n and has value zero. Choose this vertex as the anchor of H , and fill all of H with zero. If H does not meet E_n , choose any surrounding good anchor as before. Outside the holes retain the original values. The resulting field is zero on every vertex of E_n . We verify the energy bound for this choice, rather than relying on preservation by an arbitrary anchor. If an edge joins a good vertex x to a hole, its new jump is the difference of u(x) and the value at that hole's chosen rim anchor. Join these two rim vertices by the good rim path just described and telescope. The path length, the number of incident edges, and the repeated uses of a rim edge have the same polynomial diameter bounds for every choice of anchor. Applying the path inequality and then Holder gives exactly the weighted bound in Lemma 5 ([Sparse-hole extension]), with possibly a changed fixed exponent of 1+d . Edges with both endpoints in a filled hole have zero jump. Edges outside the holes retain their old jump; exterior zero--zero edges may be given weight one and contribute no energy. In particular the weighted gradient estimate holds across the actual polygon boundary. The spatial moment estimates and expected energy bound then give its squared W^ 1,p gradient bound. The local weighted square bound already proved controls its L^p norm. The new anchor also preserves the vanishing filling error. For holes of diameter at most M , join each vertex to its selected anchor by a full-lattice path of length at most CM . Such paths lie in a fixed enlargement of the hole's bounding box, and their congestion is bounded by a fixed polynomial of M . The all-edge estimate applies to the zero-extended original field, including boundary-incident edges. It makes the expected error from these holes tend to zero for each fixed M . For larger holes, |U(x)-u(x)|≤C(1+d_x) by full-lattice Lipschitzness; high diameter moments make their error tend to zero as M→∞ . These estimates also cover the exceptional event by setting U=0 there and using the established moment bounds. For a comparison with identical boundary heights, z=h-h=0 on the boundary. Its extension by zero is 2 -Lipschitz on the full lattice: edges crossing the boundary first meet a fixed boundary vertex, and exterior edges have zero difference. Apply the construction to z , or to both of its tails and subtract. All triangle vertices outside the lattice polygon have value zero, so affine interpolation vanishes there as well. Norm equivalence on the fixed lattice triangles gives the displayed global Sobolev bound. This proof uses the unbounded connected exterior, and does not apply to a hole containing an entire isolated internal component of observed vertices.