Lemma 16 ([Local paths near good boxes]). Choose a representative cut hole i(z) in every good box z . Any two cut holes in one good box, and representative holes in two nearest-neighbor good boxes, can be joined in Γ using at most C_β …

A good box contains an unblocked site, since otherwise an M -ball in it would contain six blocked sites. To connect two desired sites, first use a coordinate path of length C d through the one- or two-box neighborhood. Replace every blocked vertex of this path by a surviving nearest neighbor. Such a neighbor exists: six blocked neighbors would contradict (i). Consecutive replaced vertices have distance at most three. The imported local-connectivity result joins them through unblocked sites with bounded length and displacement. All its balls lie in the 10d -neighborhood where (i) is absent. Lift the resulting path to cut slots, using the complete adjacency of available slots across a site edge. If different slots at one site need joining, pass through a surviving neighbor of that site. The absence of (iii) means that these cut adjacencies persist for at least δ after the cut. This proves the first assertion. For the second assertion, a realization has at most two monotone mark pieces and a contact edge. Suppose one piece meets z . The portions on either side of that visit each have fewer than j_* steps: a longer portion, read in the appropriate time direction, would violate (ii). If there is a second piece, its contact endpoint is therefore within 2j_*+1 of the visit. This point lies in the tested 10d -neighborhood, and (ii) bounds the whole second piece by j_*-1 steps. If the contact edge itself meets z , apply the same argument from its endpoints. Thus every point of the realization lies within 3j_*+2<d/2 of the visit. This reasoning also applies in a lift: every monotone piece uses distinct torus marks during one period.